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Noncommutative ring

In ring theory, a noncommutative ring is an algebraic structure consisting of a nonempty set equipped with two binary operations—typically called addition and multiplication—that satisfy the standard ring axioms: addition forms an abelian group, multiplication is associative, has a multiplicative identity, and is distributive over addition, but multiplication is not required to be commutative, meaning there exist elements a and b in the ring such that ab \neq ba.^[1] This contrasts with commutative rings, where ab = ba holds for all elements a, b.^[2] Key examples of noncommutative rings include the ring of n \times n matrices over a field K (denoted M_n(K)) for n \geq 2, where matrix multiplication fails to commute in general, and the ring of quaternions \mathbb{H}, a four-dimensional division ring over the real numbers with basis \{1, i, j, k\} satisfying ij = k, ji = -k.^[3]^[4] Other significant instances arise in group rings k[G], formed by formal linear combinations of group elements with coefficients in a field k, and in operator algebras, such as the algebra of bounded linear operators on a Hilbert space.^[3] Noncommutative rings play a central role in modern algebra, extending classical results like the structure theorem for commutative rings to more general settings, and they underpin developments in representation theory (via group and algebra representations) and noncommutative geometry.^[5] Their study also yields applications in coding theory, where modules over such rings facilitate error-correcting codes, and in Lie algebra theory through enveloping algebras.^[6] Fundamental theorems, such as the Artin-Wedderburn theorem, decompose semisimple noncommutative rings into matrix rings over division rings, highlighting their structural complexity.^[5]

Definition and Basic Concepts

Definition

A noncommutative ring is a ring R in which the multiplication operation is not commutative, meaning there exist elements a, b \in R such that ab \neq ba.^[7] More precisely, a ring R is a set equipped with two binary operations, addition + and multiplication \cdot (often denoted simply by juxtaposition), satisfying the following axioms: (R, +) forms an abelian group with identity element $0 (so addition is associative and commutative, every element has an additive inverse, and a + 0 = a for all a \in R); multiplication is associative, i.e., (ab)c = a(bc) for all a, b, c \in R; and the distributive laws hold: a(b + c) = ab + ac and (a + b)c = ac + bc for all a, b, c \in R.^[8] In many contexts, particularly in noncommutative ring theory, rings are assumed to have a multiplicative identity element $1 (also called a unital ring or ring with unity), satisfying $1a = a1 = a for all a \in R, making (R, \cdot) a monoid. However, the more general notion of a rng (ring without identity) omits the requirement for a multiplicative identity, allowing multiplication to form merely a semigroup.^[7] The distributive properties are always left- and right-distributive with respect to addition, but noncommutativity implies that left and right actions may differ significantly. Central to the study of noncommutative rings are ideals, which generalize subgroups under the ring operations and are crucial for quotient constructions. A left ideal of R is an additive subgroup I \subseteq R such that r a \in I for all r \in R and a \in I; a right ideal satisfies a r \in I for all r \in R and a \in I; and a two-sided ideal (or simply ideal) is a subset that is both a left and right ideal, so r a, a r \in I for all r \in R, a \in I.^[9] For the commutative baseline, the ring of integers \mathbb{Z} under usual addition and multiplication has ideals that coincide as left, right, and two-sided, namely the principal ideals n\mathbb{Z} for n \geq 0.^[7] A concrete example of a noncommutative ring is the ring of $2 \times 2 matrices over \mathbb{R}, where matrix multiplication fails to commute in general.

Elementary Properties

A noncommutative ring R shares the same additive structure as any ring: the set R forms an abelian group under addition, with every element r \in R having an additive inverse -r such that r + (-r) = 0, the additive identity. The zero element satisfies $0 \cdot r = r \cdot 0 = 0 for all r \in R, and multiplication distributes over addition from both sides: a(b + c) = ab + ac and (b + c)a = ba + ca.^[10] Multiplication in R is associative, meaning (ab)c = a(bc) for all a, b, c \in R, but lacks commutativity, so ab \neq ba in general. This noncommutativity precludes the satisfaction of universal polynomial identities, such as those relying on ab = ba, that hold in all commutative rings.^[11] Assuming R has a multiplicative identity $1, an element u \in R is a unit if there exists v \in R such that uv = vu = 1; the set of all units forms a group under multiplication.^[10] Zero divisors are defined asymmetrically due to noncommutativity: a nonzero element a \in R is a left zero divisor if there exists nonzero b \in R with ab = 0, and a right zero divisor if ba = 0 for some nonzero b. A ring with no left (respectively, right) zero divisors is called a left (right) domain; for instance, division rings like the quaternions have no zero divisors and thus are both left and right domains.^[11] The center of R, denoted Z(R) = \{ z \in R \mid zr = rz \ \forall r \in R \}, consists of all elements that commute with every element of R; it forms the largest commutative subring of R, containing the scalars if R is an algebra over a field.^[10] In matrix rings over a field, for example, the center is precisely the scalar matrices.^[11] A two-sided ideal I of R is an additive subgroup such that rI \subseteq I and Ir \subseteq I for all r \in R, making it "normal" under left and right multiplication by ring elements; the zero ideal \{0\} and R itself are always two-sided ideals. The kernel of any ring homomorphism \phi: R \to S is \ker \phi = \{ r \in R \mid \phi(r) = 0 \}, which forms a two-sided ideal of R.^[10]

Historical Development

Origins and Early Contributions

The origins of noncommutative ring theory trace back to the mid-19th century, when mathematicians sought to extend the successful framework of complex numbers to higher dimensions, particularly for representing three-dimensional rotations. In 1843, William Rowan Hamilton discovered quaternions while walking along the Royal Canal in Dublin on October 16, en route to a meeting of the Royal Irish Academy; motivated by the failure of commutativity in vector multiplications for 3D space, he realized that a four-dimensional algebra was necessary, introducing the first explicit example of a noncommutative algebraic structure.^[12] This breakthrough was so profound that Hamilton immediately carved the fundamental relations on Brougham Bridge and recorded the insight in his notebook that same day.^[13] Building on Hamilton's work, Arthur Cayley advanced the study of noncommutative structures through his development of matrix algebras in 1858, where he formalized matrices as entities that could be added and multiplied, explicitly recognizing their noncommutative multiplication as a key feature akin to that in hypercomplex numbers like quaternions.^[14] Cayley's seminal paper, "A Memoir on the Theory of Matrices," treated matrices as single quantities under composition, laying groundwork for associative algebras and linking them to Hamilton's quaternions, which he had encountered earlier through lectures and correspondence.^[15] Hamilton himself continued refining quaternions in subsequent publications, emphasizing their noncommutative nature as essential for geometric applications beyond commutative complex numbers.^[12] Throughout the 19th century, algebraic developments were predominantly focused on commutative cases, influenced by number-theoretic pursuits such as those of Richard Dedekind and Leopold Kronecker, who introduced ideals and orders in commutative rings to generalize arithmetic in algebraic number fields, with little initial attention to noncommutativity.^[16] Dedekind's 1871 work on ideals and Kronecker's emphasis on polynomial rings prioritized commutative structures for solving problems like Fermat's Last Theorem, sidelining hypercomplex extensions.^[17] A pivotal clarification came in 1877 with Ferdinand Georg Frobenius's theorem, which classified all finite-dimensional associative division algebras over the real numbers as the reals, complexes, or quaternions, underscoring the rarity and significance of noncommutative examples like Hamilton's invention.^[18]

Modern Foundations

The early 20th century marked a pivotal shift toward axiomatic approaches in algebra, with Emmy Noether's work in the 1920s profoundly influencing the abstract treatment of rings, including noncommutative structures, by emphasizing ideals and modules as fundamental tools for generalization beyond specific number systems.^[19] Concurrently, Leonard Eugene Dickson and Joseph Wedderburn advanced the classification of finite-dimensional algebras over fields during the 1900s and 1910s, laying groundwork for understanding noncommutative examples like matrix algebras and division rings.^[20] Wedderburn's seminal 1905 paper established key results on semisimple algebras, proving that finite division rings are commutative and providing an initial decomposition into matrix components over division rings. Key milestones in the 1920s included Emil Artin's formalization of general ring theory, extending beyond hypercomplex numbers to arbitrary associative rings without commutativity assumptions.^[21] Artin's 1927 contributions specifically addressed noncommutative polynomials and their ideals, bridging polynomial rings to broader algebraic structures and influencing subsequent developments in representation theory. These efforts solidified the axiomatic foundation, distinguishing noncommutative rings from their commutative counterparts by highlighting the need for left and right ideals. Advancements in the mid-20th century included Nathan Jacobson's structure theory in the 1940s and 1950s, which generalized Artin-Wedderburn results to rings without finiteness conditions, introducing concepts like primitive rings and density arguments—such as the Jacobson density theorem serving as a cornerstone for module faithfulness. Jacobson's 1943 book The Theory of Rings synthesized these ideas, providing a comprehensive framework for noncommutative structures.^[22] The 1950s saw further extension through Henri Cartan and Samuel Eilenberg's Homological Algebra (1956), which developed module theory for noncommutative rings, incorporating derived functors and resolutions to analyze extensions and cohomology.^[23] Representation theory from Lie groups also impacted this era, informing ring decompositions via group algebras and enveloping algebras.

Examples

Matrix Rings

Matrix rings provide a fundamental class of noncommutative rings, constructed as follows: for a ring D (typically a division ring or a commutative ring) and a positive integer n, the ring M_n(D) consists of all n \times n matrices with entries in D, equipped with componentwise addition and the standard matrix multiplication (AB)_{ij} = \sum_{k=1}^n a_{ik} b_{kj}.^[11] This multiplication operation ensures that M_n(D) forms a ring, and when D is non-trivial, the ring is generally noncommutative, as illustrated by the generic $2 \times 2 matrices A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} and B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} over a field, where AB \neq BA.^[24] The full matrix ring M_n(D) over a division ring D exhibits key structural properties: it is both simple (possessing no non-trivial two-sided ideals) and semisimple (as a finite direct sum of simple modules over itself).^[25] Furthermore, as a left vector space over its center (which coincides with the center of D), M_n(D) has dimension n^2.^[11] An important equivalence relation in ring theory is that M_n(D) is Morita equivalent to D, meaning their module categories are isomorphic, which preserves many homological properties between the rings.^[26] A concrete example is M_2(\mathbb{R}), the ring of $2 \times 2 real matrices, which is noncommutative and serves as the prototypical finite-dimensional noncommutative algebra over the reals; its elements correspond to linear transformations of \mathbb{R}^2, with multiplication reflecting composition of these transformations.^[24] More generally, for any commutative ring D, M_n(D) models endomorphisms of free D-modules of rank n, highlighting the role of matrix rings in representation theory and linear algebra over rings.^[11]

Division Rings and Quaternions

A division ring, also known as a skew field, is a ring with unity in which every nonzero element has a two-sided multiplicative inverse.^[27] Unlike fields, division rings need not have commutative multiplication, allowing for noncommutative examples where ab ≠ ba for some a, b.^[1] The quaternions, discovered by William Rowan Hamilton in 1843, provide a fundamental example of a noncommutative division ring.^[28] Hamilton's quaternions \mathbb{H} consist of elements of the form a + bi + cj + dk where a, b, c, d \in \mathbb{R} and i, j, k satisfy i^2 = j^2 = k^2 = -1, ij = k, and ji = -k.^[29] As a vector space over \mathbb{R}, \mathbb{H} has dimension 4 with basis \{1, i, j, k\}.^[29] The center of \mathbb{H}, the set of elements commuting with all others, is precisely \mathbb{R}.^[29] Quaternion multiplication can be expressed in vector notation by identifying pure quaternions (those with zero real part) with vectors in \mathbb{R}^3. For q = a + \mathbf{u} and q' = b + \mathbf{v} where a, b \in \mathbb{R} and \mathbf{u}, \mathbf{v} \in \mathbb{R}^3 (viewed as pure quaternions), the product is

qq' = ab - \langle \mathbf{u}, \mathbf{v} \rangle + a\mathbf{v} + b\mathbf{u} + \mathbf{u} \times \mathbf{v},

with \langle \cdot, \cdot \rangle the dot product and \times the cross product.^[29] This formula highlights the noncommutativity, as the cross product term \mathbf{u} \times \mathbf{v} = -\mathbf{v} \times \mathbf{u} changes sign under reversal. Every nonzero quaternion has a two-sided inverse, confirming \mathbb{H} is a division ring.^[29] More broadly, quaternions exemplify finite-dimensional division algebras over a field, which are associative algebras where nonzero elements are invertible. Over the reals, Ferdinand Georg Frobenius proved in 1877 that the only such algebras (up to isomorphism) are \mathbb{R}, \mathbb{C}, and \mathbb{H}. This classification underscores the rarity of noncommutative examples, with \mathbb{H} being the unique 4-dimensional case. For finite division rings, Wedderburn's little theorem (1905) further implies they must be commutative and thus fields.^[30]

Group Rings and Other Constructions

One important construction of noncommutative rings arises from group rings, which combine the structure of a group with a coefficient ring. Given a ring R (typically commutative) and a group G, the group ring RG consists of all formal finite sums \sum_{g \in G} r_g g where r_g \in R and only finitely many r_g are nonzero.^[31] The addition in RG is componentwise, and the multiplication is defined by the convolution formula:

\left( \sum_{g \in G} r_g g \right) \left( \sum_{h \in G} s_h h \right) = \sum_{m \in G} \left( \sum_{g k = m} r_g s_k \right) m,

extended linearly from the group multiplication in G.^[31] If G is nonabelian, then RG is generally noncommutative, as the group elements do not commute.^[31] A key property of the group ring RG is the augmentation ideal, which is the kernel of the augmentation map \varepsilon: RG \to R defined by \varepsilon\left( \sum r_g g \right) = \sum r_g.^[31] This ideal, often denoted \Delta(RG) or I_G, is generated by elements of the form g - 1 for g \in G.^[31] For example, consider the group ring \mathbb{Z}S_3, where S_3 is the nonabelian symmetric group on three letters. Elements are integer linear combinations of the six group elements, such as $5(12) + 3(13) + 2(123), and the ring is noncommutative due to the relations in S_3; the augmentation ideal consists of sums with coefficients totaling zero.^[32] Beyond group rings, other constructions highlight noncommutativity through generators without imposed relations or specific commutation rules. The free algebra k\langle x, y \rangle over a field k, also denoted k\{x, y\}, is the associative algebra generated by noncommuting indeterminates x and y with no further relations; its elements are finite sums of terms like a x^{i_1} y^{j_1} x^{i_2} y^{j_2} \cdots, where the variables do not commute (xy \neq yx).^[5] This provides the "freest" noncommutative structure extending polynomial rings. Another prominent example is the Weyl algebra, which models differential operators. The first Weyl algebra A_1(k) over a field k of characteristic zero is generated by elements x (multiplication by the indeterminate) and \partial (formal differentiation) satisfying the commutation relation [\partial, x] = \partial x - x \partial = 1.^[33] It can be realized as the ring of k-linear endomorphisms of the polynomial ring k generated by multiplication by x and differentiation, making it a noncommutative domain that is simple but not a matrix ring over a division ring.^[33]

Differences from Commutative Rings

Structural Distinctions

In noncommutative rings, the lack of commutativity in multiplication leads to significant differences in algebraic structures compared to their commutative counterparts. For instance, polynomials over noncommutative rings, such as the free algebra k\langle x, y \rangle generated by noncommuting indeterminates over a field k, do not admit unique factorization. A classic example is the element x + xyx, which factors non-uniquely as x(1 + yx) = (1 + xy)x, where the factors are distinct and irreducible. This contrasts sharply with commutative polynomial rings, where unique factorization holds under mild conditions, highlighting how noncommutativity disrupts the usual arithmetic properties. Ideals in noncommutative rings require careful distinction between left ideals, right ideals, and two-sided ideals. A left ideal I of a ring R is a subgroup closed under left multiplication by elements of R (i.e., r i \in I for all r \in R, i \in I), while right ideals satisfy i r \in I, and two-sided ideals satisfy both. In commutative rings, these categories coincide, but in noncommutative rings, they generally differ; for example, in the matrix ring M_2(k) over a field k, the set of matrices with zero first row forms a left ideal but not a right ideal. Moreover, the presence of only two-sided ideals does not imply commutativity, as M_n(k) for n > 1 has no nontrivial two-sided ideals despite being noncommutative. Ring homomorphisms between noncommutative rings preserve the additive and multiplicative structures but do not enforce commutativity in the image. Specifically, a homomorphism \phi: R \to S satisfies \phi(a + b) = \phi(a) + \phi(b) and \phi(ab) = \phi(a)\phi(b), so if ab = ba in R, then \phi(a)\phi(b) = \phi(b)\phi(a) in S, but noncommuting elements in R may map to commuting ones. For instance, the projection \phi: M_2(k) \to k sending a matrix to its (1,1)-entry is a ring homomorphism onto the commutative ring k, collapsing the noncommutativity. Conversely, noncommutative images arise naturally, such as the identity map on M_2(k) itself or the canonical embedding of the quaternion algebra into larger division rings. Subrings of noncommutative rings inherit the ring operations but need not be commutative, even though the center—the subring of elements commuting with everything—is always commutative. For example, in the ring of $2 \times 2 upper triangular matrices over k, the full subring of all such matrices is noncommutative, while its center consists of scalar matrices, which is commutative. This illustrates how noncommutativity can permeate substructures without affecting the central elements. Zero divisors are more prevalent in noncommutative settings, as in matrix rings where nonzero elements can multiply to zero.

Module and Ideal Theory Variations

In noncommutative rings, the theory of modules must account for the side of the ring action, leading to distinct categories of left and right modules. A left R-module _RM is an abelian group equipped with a map R \times M \to M, (r,m) \mapsto rm, satisfying distributivity and associativity: r(m_1 + m_2) = rm_1 + rm_2, (r_1 + r_2)m = r_1 m + r_2 m, and (r_1 r_2)m = r_1 (r_2 m). Dually, a right R-module MR has the action on the right, with m(r_1 r_2) = (m r_1) r_2. Noncommutativity prevents the left and right actions from being interchanged freely, resulting in asymmetric structures where properties like projectivity or injectivity may differ between left and right modules over the same ring.^[34]^[5] Bimodules extend this framework by incorporating compatible actions from both sides. An R-bimodule _RM_R is a left R-module that is also a right R-module, with the actions commuting: (rm)s = r(ms) for all r,s \in R, m \in M. This compatibility ensures that the bimodule supports two-sided ring operations, which is crucial for constructions like tensor products over noncommutative rings or Morita equivalences between categories of left and right modules. In contrast to commutative rings, where left and right modules coincide, bimodules over noncommutative rings highlight the need to track both actions explicitly, as failure of commutativity can lead to non-isomorphic left and right module categories.^[5]^[34] Ideal theory in noncommutative rings further diverges from the commutative case due to the distinction between one-sided and two-sided ideals. A left ideal I satisfies RI \subseteq I, a right ideal J satisfies JR \subseteq J, and a two-sided ideal satisfies both. Ascending and descending chains of ideals are analyzed separately for left, right, and two-sided cases; noncommutativity implies that satisfaction of the ascending chain condition (ACC) on left ideals does not guarantee it for right ideals, preventing a unified Noetherian property. For instance, a ring may be left Noetherian (every left ideal is finitely generated, equivalent to ACC on left submodules of _RR) but fail to be right Noetherian, and similarly for the descending chain condition defining Artinian rings. This side-specific behavior underscores the noncommensurability of chain conditions in noncommutative settings.^[34] The Jacobson radical J(R) captures essential structural information about these variations. Defined as the intersection of all maximal left ideals of R (equivalently, all maximal right ideals, as the two coincide), J(R) consists of elements that act as zero divisors on every simple left R-module, or more precisely, the intersection of the annihilators of all simple left modules. In noncommutative rings, J(R) contains every nil left ideal and, under additional hypotheses like Artinianity, is itself nilpotent, meaning some power J(R)^n = 0. Unlike in commutative rings, where J(R) is the nilradical, the noncommutative version emphasizes quasi-regular elements and plays a key role in lifting idempotents modulo J(R). Nakayama's lemma extends to this context, applying to finitely generated modules over rings with nilpotent Jacobson radical.^[35]^[34] A representative example illustrating these variations is the ring of n \times n matrices M_n(k) over a field k. As a noncommutative simple Artinian ring, M_n(k) has only trivial two-sided ideals: \{0\} and itself. However, it possesses numerous proper one-sided ideals; for instance, the set of matrices with the first column zero forms a maximal left ideal, and dually for right ideals. The Jacobson radical J(M_n(k)) = \{0\}, reflecting its semisimplicity, yet the abundance of one-sided ideals demonstrates how noncommutativity proliferates left/right structures while restricting two-sided ones. Left modules over M_n(k) are equivalent to vector spaces over k via Morita equivalence, but the explicit side distinctions persist in ideal chains and bimodule constructions.^[34]^[5]

Classes of Noncommutative Rings

Simple Rings

A simple ring is a nonzero ring R that possesses no two-sided ideals other than the zero ideal and R itself.^[34] This property underscores the indecomposability of simple rings with respect to two-sided ideal structure, distinguishing them as the "atoms" in the lattice of ring ideals.^[5] Basic examples include division rings, which have no proper two-sided ideals by virtue of every nonzero element being invertible, and full matrix rings M_n(D) over a division ring D, where the two-sided ideals correspond precisely to those of D.^[34] Simple rings exhibit key module-theoretic properties: they are primitive as left (or right) modules over themselves, meaning the regular module _RR (or R_R) admits a faithful simple submodule whose annihilator is zero, owing to the absence of nontrivial two-sided ideals.^[5] This primitivity ensures that simple rings faithfully act on their simple modules via irreducible representations.^[5] In the broader context of semisimple rings, the Wedderburn–Artin theorem decomposes them into direct sums of simple rings, highlighting the role of simples as building blocks (detailed further in the Key Theorems section).^[34] For the finite-dimensional case over a field—equivalently, Artinian simple rings—the structure is particularly rigid: every such ring is isomorphic to a matrix ring M_n(D) over a division ring D.^[34] This characterization follows from the Artin–Wedderburn theorem but is previewed here without proof, emphasizing that Artinian simples are precisely the matrix rings over their unique minimal division subrings.^[5] Representative examples abound in linear algebra settings: the ring of n \times n matrices M_n(\mathbb{R}) over the reals is a simple noncommutative ring, as are endomorphism rings \mathrm{End}_D(V) of finite-dimensional vector spaces V over a division ring D, which are isomorphic to M_n(D) where n = \dim_D V.^[34] These constructions illustrate how simple rings arise naturally from representations of algebras on vector spaces.^[5]

Division Rings

A division ring, also known as a skew field, is a nonzero ring D with multiplicative identity in which every nonzero element admits a two-sided multiplicative inverse.^[36] This invertibility ensures that D has no zero divisors, making it an integral domain in the noncommutative sense. The center Z(D) = \{ z \in D \mid zd = dz \ \forall d \in D \} forms a commutative subring that is itself a field, serving as the scalar field over which D can be viewed as an algebra.^[36] Division rings thus generalize fields by relaxing the commutativity of multiplication while preserving the ability to divide by nonzero elements. When a division ring D is finite-dimensional as a vector space over its center Z(D), it is classified up to isomorphism by the Brauer group \mathrm{Br}(Z(D)), which parameterizes central simple algebras over the field Z(D) via Morita equivalence.^[37] Specifically, the class [D] in \mathrm{Br}(Z(D)) has order dividing the reduced degree (or index) of D, and the dimension \dim_{Z(D)} D = n^2 for some integer n, where n is the index.^[37] This classification highlights division rings as the "indecomposable" building blocks of more general central simple algebras, with the Brauer group capturing obstructions to splitting such algebras into matrix rings over the center. Infinite-dimensional examples of division rings abound beyond finite cases. Rational function skew fields arise as universal localizations of Ore domains, such as the skew polynomial ring k\langle x \rangle[y; \sigma] over a field k, where \sigma is a nontrivial automorphism of k\langle x \rangle, yielding a division ring of fractions containing noncommuting rational expressions.^[38] Free fields provide another construction: the free field on m noncommuting indeterminates over a field k is the division subring of the free associative algebra k\langle x_1, \dots, x_m \rangle generated by localizing at nonzero elements, resulting in an infinite-dimensional division ring with no relations beyond noncommutativity. These examples illustrate the richness of noncommutative rational structures. Division rings exhibit strong structural properties, including being both left and right Artinian, as they possess descending chains of ideals that terminate (in fact, the only ideals are \{0\} and D itself).^[36] They satisfy the left and right Ore conditions with respect to the multiplicative set of nonzero elements, allowing them to serve as quotient rings for suitable subrings, though as maximal localizations they embed Ore domains universally.^[39] A key result is Wedderburn's little theorem, which states that every finite division ring is commutative and hence a field; this was originally proved using character theory on the unit group.^[40] As noted earlier, the real quaternions exemplify a noncommutative finite-dimensional division ring over \mathbb{R}.

Primitive Rings

A left primitive ring is defined as a ring R that possesses a faithful simple left R-module M, meaning the annihilator of M is zero and M has no proper nontrivial submodules.^[41] This condition implies that R embeds densely into the endomorphism ring \operatorname{End}_D(M), where D = \operatorname{End}_R(M) is a division ring. Equivalently, R as a left module over itself admits a simple submodule that is faithful. A ring is right primitive if it has a faithful simple right module, and some rings exhibit primitivity on one side but not the other. Simple rings are primitive, as any nonzero simple module over a simple ring is faithful, given that the annihilator of any nonzero module is a proper ideal, hence zero.^[41] Representative examples include matrix rings M_n(D) over a division ring D, where the natural left module of column vectors is simple and faithful.^[42] Another example is the ring of all linear endomorphisms of an infinite-dimensional vector space over a division ring, which is left and right primitive but not simple, as it contains proper ideals consisting of finite-rank operators.^[41] In a primitive ring, the left socle—the sum of all simple left submodules of _RR—is nonzero, containing the faithful simple submodule as a direct summand.^[43] This socle is uniform, meaning any two nonzero submodules have nonzero intersection, reflecting the module-theoretic structure induced by the faithful simple module. The Jacobson density theorem provides a key structural description for primitive rings, characterizing their action on the faithful simple module.

Semisimple Rings

A semisimple ring is defined as an Artinian ring whose Jacobson radical is zero. This condition ensures that the ring has no nonzero nilpotent ideals that are "large" in the sense of intersecting every maximal ideal. Equivalently, a ring is semisimple if and only if it is semisimple as a module over itself, meaning the regular module decomposes as a direct sum of simple submodules. In the noncommutative setting, this definition captures rings where the module category is particularly well-behaved, with every module admitting a composition series of finite length. Semisimple rings possess several key structural properties. They are finite direct sums of simple Artinian rings, and the decomposition arises from the existence of central idempotents—idempotent elements in the center of the ring that are orthogonal and sum to the identity. These central idempotents generate the two-sided ideals, allowing the ring to split into indecomposable components known as Wedderburn components (as detailed in the Artin–Wedderburn theorem). Moreover, every left (or right) module over a semisimple ring has finite length, reflecting the Artinian nature and the absence of the Jacobson radical, which would otherwise prevent such uniform finite dimensionality.^[34] Representative examples of semisimple rings include finite direct products of full matrix rings over division rings, such as M_n(D) where D is a division ring like the quaternions, or products like M_{k}(F) \times M_{m}(D) for fields F and division rings D. In these cases, the ring as a module over itself is a direct sum of simple modules corresponding to the matrix components, each of finite length equal to the matrix size times the dimension over the division ring. Such structures highlight how semisimple rings generalize commutative semisimple rings, which are merely finite products of fields, to the noncommutative realm.

Key Theorems

Wedderburn–Artin Theorem

The Wedderburn–Artin theorem provides a complete structure theorem for semisimple Artinian rings, stating that every such ring R is isomorphic to a finite direct product of full matrix rings over division rings:

R \cong \bigoplus_{i=1}^r M_{n_i}(D_i),

where each D_i is a division ring, each n_i is a positive integer, and r is finite. This decomposition classifies the ring explicitly, showing that semisimple Artinian rings are precisely the finite direct sums of matrix rings over division rings. For the simple case, where r = 1, the theorem asserts that a simple Artinian ring is isomorphic to M_n(D) for some division ring D and integer n \geq 1. A standard proof begins by leveraging the Artinian condition to find primitive idempotents. In a semisimple Artinian ring, every left ideal contains a minimal left ideal, and by Brauer's lemma, every nonzero left ideal contains a nonzero idempotent e. For a primitive idempotent e, the left ideal Re is minimal, and eRe is a division ring D. The Peirce decomposition relative to e splits R = eRe \oplus eR(1-e) \oplus (1-e)Re \oplus (1-e)R(1-e), but focusing on the simple component generated by Re, one shows R decomposes into a direct sum of simple rings via maximal orthogonal idempotents. For each simple summand S, the minimal left ideals are isomorphic as right S-modules, forming a right vector space V over D = \text{End}_S(V), and S \cong \text{End}_D(V) \cong M_n(D) by the density theorem for endomorphisms, with n = \dim_D V. The Artinian property ensures only finitely many such summands.^[44] Important corollaries follow directly. Wedderburn's little theorem, a special case, states that every finite division ring is commutative (hence a field), obtained by applying the structure theorem to the Artinian division ring itself, yielding D \cong M_n(D') for some division ring D', which forces n=1 by finiteness. Additionally, the theorem implies finite-dimensional semisimple algebras over fields have well-defined dimensions, with \dim_k R = \sum_i n_i^2 \dim_k D_i. The decomposition is unique up to isomorphism of the division rings D_i, the integers n_i, and permutation of the factors.

Jacobson Density Theorem

The Jacobson density theorem provides a fundamental characterization of primitive rings in terms of their action on faithful simple modules. Specifically, let R be a left primitive ring with a faithful simple left R-module V. Let D = \operatorname{End}_R(V)^{\mathrm{op}}, which is a division ring by Schur's lemma. Then R embeds into \operatorname{End}_D(V) via the map sending r \in R to the D-linear map v \mapsto r v, and this image is dense in \operatorname{End}_D(V). Density here means that for any finite set of elements x_1, \dots, x_n \in V that are linearly independent over D and any y_1, \dots, y_n \in V, there exists r \in R such that r x_i = y_i for all i = 1, \dots, n.^[45] The proof relies on the double centralizer theorem, which asserts that the centralizer in \operatorname{End}(V) of the image of R is precisely D, and conversely, the centralizer of D contains R densely. To establish density, proceed by induction on n. For n=1, since V is simple and faithful, any nonzero x_1 generates a submodule, and there exists r mapping it to y_1 by irreducibility. For n > 1, construct elements in R that act as "projections" onto the spans of the x_i, using the faithfulness and simplicity to ensure linear independence is preserved, and apply the inductive hypothesis to the quotients or complements. This leverages the fact that annihilators of subsets are controlled by the module's simplicity.^[45]^[46] This theorem classifies primitive rings by realizing them as dense subrings of endomorphism rings over division rings, extending the structure theory beyond finite-dimensional cases. In particular, it implies that primitive rings with minimal ideals are simple, as their actions mimic those of full matrix rings in the dense limit. For the finite-dimensional case, if \dim_D V < \infty, the density embedding becomes an isomorphism, so R \cong \operatorname{End}_D(V) \cong M_k(D) for some k, reducing to matrix rings over division rings.^[45]^[47]

Goldie's Theorem

Goldie's theorem is a cornerstone result in the structure theory of noncommutative rings, establishing conditions under which prime rings admit a classical quotient ring with desirable properties. Specifically, let R be a prime ring satisfying the ascending chain condition (ACC) on left annihilators. Then R has finite left uniform Goldie dimension, and the classical left quotient ring Q_l(R) exists and is a simple Artinian ring, with the natural embedding R \hookrightarrow Q_l(R) making R a dense subring in the sense of module theory.^[48] The proof proceeds by first establishing that such rings satisfy the left Ore condition on regular elements, allowing the construction of the classical quotient ring as a localization. Central to this are injective hulls of modules over R: the injective hull E(_R R) of the left regular module is shown to be a finite direct sum of uniform injective modules, reflecting the finite Goldie dimension. Uniform modules—those in which every nonzero submodule is essential—play a key role, as the ACC on annihilators ensures no infinite ascending chains of annihilators, leading to the finiteness of direct sum decompositions of essential ideals. This implies the Ore condition, and further analysis shows that Q_l(R) satisfies the descending chain condition on left ideals, hence is Artinian, and its simplicity follows from the primeness of R.^[48] Important corollaries include a characterization of simple Artinian rings: a ring is simple Artinian if and only if it is prime and satisfies the ACC on (left and right) annihilators, as such rings automatically have finite uniform dimension and their own classical quotient ring is themselves. Another corollary affirms that the uniform left Goldie dimension is well-defined and finite under these hypotheses, providing a measure of the "size" of the ring analogous to Krull dimension in commutative algebra. The left Goldie dimension, or uniform dimension, of R is formally defined as the maximum integer n such that there exist nonzero left ideals I_1, \dots, I_n with

_R R \cong I_1 \oplus \cdots \oplus I_n

as left R-modules, or more generally the supremum over all such direct sum lengths for submodules of _R R. Each I_i is uniform in this maximal decomposition.^[48]

Nakayama's Lemma

Nakayama's lemma is a fundamental result in module theory that provides conditions under which a finitely generated module over a ring must vanish, based on its interaction with the ring's radical. In the classical setting, for a commutative local ring R with maximal ideal \mathfrak{m}, the lemma states that if M is a finitely generated R-module satisfying \mathfrak{m}M = M, then M = 0. This version, originally due to Krull and generalized by Nakayama, highlights how "small" perturbations by the maximal ideal cannot generate the entire module unless it is trivial. The result extends naturally to the more general case where \mathfrak{m} is replaced by any ideal contained in the Jacobson radical, ensuring the conclusion holds across a broader class of rings. In the noncommutative setting, the lemma adapts to arbitrary (associative, unital) rings R by incorporating the Jacobson radical J(R), the intersection of all primitive ideals of R. For a finitely generated left R-module M, if J(R)M = M, then M = 0. This formulation preserves the essence of the commutative case but accounts for the one-sided nature of modules over noncommutative rings, where left and right versions may differ slightly but the core implication remains. A related consequence is that if N \subseteq M is a submodule such that N + J(R)M = M, then N = M, providing a criterion for submodules to complement the action of the radical. The proof of the noncommutative version proceeds by induction on the minimal number of generators of M. Assume M \neq 0 is a counterexample with minimal generating set \{m_1, \dots, m_n\} where n \geq 1. Since J(R)M = M, we have m_n = \sum_{i=1}^n r_i m_i for some r_i \in J(R). Rearranging gives (1 - r_n)m_n = \sum_{i=1}^{n-1} r_i m_i. A key property of the Jacobson radical ensures that there exists u \in R such that u(1 - r_n) = 1, because R(1 - r_n) = R for any r_n \in J(R). Multiplying the equation on the left by u yields m_n = u \sum_{i=1}^{n-1} r_i m_i, expressing m_n as an R-linear combination of m_1, \dots, m_{n-1}, which contradicts the minimality of the generating set. Thus, no such nontrivial M exists. The second form follows by applying the first to the quotient M/N.^[49] This lemma finds essential applications in noncommutative ring theory, particularly in determining minimal generating sets and complements for modules. A set of elements generates M if and only if their images generate M / J(R)M, allowing reduction to the semisimple quotient ring R / J(R). Additionally, it guarantees the existence of complements: for a short exact sequence $0 \to N \to M \to M/N \to 0 with M finitely generated, if M/N is generated by fewer elements than a minimal generating set of M, then the sequence splits, facilitating the study of module decompositions over rings with nontrivial radicals. These tools are crucial for computations in representation theory and the structure of algebras.

Advanced Structures and Equivalences

Noncommutative Localization

In noncommutative ring theory, the localization of a ring R at a multiplicative subset S \subseteq R is defined when S satisfies the Ore conditions, which ensure the existence of a suitable ring of fractions. The right Ore condition requires that for all s \in S and r \in R, there exist s' \in S and r' \in R such that s r' = r s'; the left Ore condition is the symmetric requirement r' s = s' r. These conditions, introduced by Øystein Ore, allow the construction of a quotient ring where elements of S become invertible, generalizing the classical localization in commutative algebra.^[50]^[51] The Ore localization S^{-1}R is constructed as the quotient of the set of pairs (r, s) with r \in R and s \in S by an equivalence relation: (r, s) \sim (r', s') if there exists u \in S such that u (r s' - r' s) = 0. Addition and multiplication are defined by (r, s) + (r', s') = (r s' + r' s, s s') and (r, s) (r', s') = (r r', s s'), respectively, making S^{-1}R a ring with a canonical homomorphism \iota: R \to S^{-1}R given by r \mapsto (r, 1). This construction yields a left (or right) R-module structure, and if both Ore conditions hold, S^{-1}R is unambiguous up to isomorphism.^[11]^[51] A key property of Ore localization is its universal property: for any ring homomorphism f: R \to T such that f(s) is invertible in T for all s \in S, there exists a unique ring homomorphism \overline{f}: S^{-1}R \to T extending f, satisfying \overline{f} \circ \iota = f. In the case where R is a domain and S = R \setminus \{0\} satisfies the Ore conditions, S^{-1}R is a skew field of fractions, embedding R into a division ring.^[50]^[11] Representative examples illustrate these concepts. The ring of Laurent polynomials k[x, x^{-1}] over a field k arises as the localization of the polynomial ring k at the multiplicative set S = \{x^n \mid n \geq 0\}, where the Ore conditions hold since x^n r = r x^n for monomials r. Similarly, the first Weyl algebra A_1(k) = k\langle x, \partial \rangle / (\partial x - x \partial - 1) admits localizations at sets generated by monomials in x and \partial, yielding rings where these operators become invertible while preserving the relation.^[51]^[52]

Morita Equivalence

In ring theory, two rings R and S are said to be Morita equivalent if there exists an equivalence of categories between the category of left R-modules, denoted \Mod-R, and the category of left S-modules, \Mod-S. This equivalence is typically induced by a pair of adjoint functors that are inverses to each other, such as F: \Mod-R \to \Mod-S and G: \Mod-S \to \Mod-R satisfying F \dashv G and FG \cong \mathrm{id}_{\Mod-S}, GF \cong \mathrm{id}_{\Mod-R}.^[53] An equivalent formulation states that R and S are Morita equivalent if there exists an (R, S)-bimodule _R P_S that is a progenerator as an R-module (finitely generated projective and generates \Mod-R) such that S \cong \End_R(P)^{\mathrm{op}}, the opposite ring of the endomorphism ring of P.^[54] A key criterion for Morita equivalence involves the existence of balanced bimodules. Specifically, rings R and S are Morita equivalent if there are an (R, S)-bimodule M and an (S, R)-bimodule N such that the natural homomorphisms

M \otimes_S N \to R, \quad N \otimes_R M \to S

are isomorphisms, and the canonical associativity maps induce isomorphisms between the multiple tensor products. Equivalently, the functor F(M) = M \otimes_R -\colon [\Mod-R](/page/Module) \to [\Mod-S](/page/Module) is fully faithful and dense (every S-module is a quotient of some M \otimes_R V for V \in \Mod-R). These conditions ensure the module categories are isomorphic while allowing the rings themselves to differ substantially.^[54]^[53] Classic examples illustrate this equivalence. For any ring R and positive integer n, the matrix ring M_n(R) is Morita equivalent to R, with the standard bimodule being the row space R^n, which serves as a progenerator over R and yields \End_R(R^n) \cong M_n(R). This equivalence arises because the category of M_n(R)-modules is isomorphic to \Mod-R via the forgetful functor that views column vectors as R-modules. Another example occurs with group rings: over a field k, the group algebra kG of a finite group G is Morita equivalent to a direct product of matrix rings over division rings corresponding to the irreducible representations of G, provided the representations satisfy the conditions of the Artin-Wedderburn theorem for semisimple algebras.^[53]^[54] Morita equivalence preserves numerous ring-theoretic properties tied to module categories, such as the Jacobson radical (the radical corresponds under the equivalence, ensuring quasi-regularity is maintained) and the lengths of modules (composition series lengths are invariant). It also preserves global dimension, Artinian and Noetherian conditions, and the structure of simple modules up to isomorphism. However, while the centers of Morita equivalent rings are isomorphic (as the center acts centrally on modules), the equivalence does not induce a ring isomorphism on the centers themselves, allowing non-isomorphic rings with identical centers to be equivalent. This distinction highlights how Morita equivalence captures representational similarity rather than structural identity.^[54]^[53]

Ore Conditions and Classical Rings

In noncommutative ring theory, the Ore conditions provide necessary and sufficient criteria for constructing localizations analogous to fields of fractions in the commutative case. A multiplicatively closed subset S of a ring R satisfies the right Ore condition if, for every a \in R and s \in S, there exist b \in R and s' \in S such that a s' = b s, or equivalently, a S \cap R s \neq \emptyset.^[34] The left Ore condition is defined dually: for every a \in R and s \in S, there exist b \in R and s' \in S such that s' a = s b, or S a \cap s R \neq \emptyset.^[34] These conditions ensure that the localization R S^{-1}, consisting of right fractions r s^{-1} with r \in R, s \in S, can be well-defined as a ring, with equality of fractions determined unambiguously by the Ore property. If S satisfies both left and right Ore conditions, the left and right localizations coincide. The classical quotient ring arises when S is the set of regular elements of R, typically the nonzero-divisors in a domain. If this set satisfies the (say) right Ore condition, the resulting localization Q = R S^{-1} is the classical right quotient ring of R, embedding R as a subring and serving as a maximal such localization.^[34] For domains where the set of nonzero elements satisfies both Ore conditions, Q is a division ring, unique up to isomorphism.^[55] Ore introduced these constructions in the context of noncommutative polynomials, showing they enable division algorithms and Euclidean-like properties in suitable extensions. Prime rings admitting a classical quotient division ring exhibit strong structural properties, including the ascending chain condition (ACC) on right annihilators. This ACC ensures no infinite ascending chains of right annihilator ideals, linking directly to the Goldie dimension, which measures the uniformity of the ring as the number of minimal direct summands in essential extensions. Such rings have finite uniform (Goldie) dimension, providing a noncommutative analogue of Krull dimension in prime ideals. A representative example is the free algebra k\langle x_1, \dots, x_n \rangle over a field k in noncommuting variables, which is a domain where all nonzero elements are regular and satisfy both Ore conditions. Its classical quotient ring is the free field, a division ring embodying the universal property for rational expressions in noncommuting indeterminates.

Brauer Groups

The Brauer group of a field k, denoted \mathrm{Br}(k), consists of the set of similarity classes of central simple k-algebras, where two such algebras A and B are similar if A \otimes_k M_m(k) \cong B \otimes_k M_n(k) for some positive integers m and n.^[56] The group operation on \mathrm{Br}(k) is induced by the tensor product of algebras over k, with the identity element given by the similarity class of matrix algebras over k and the inverse of a class [A] given by the class of the opposite algebra A^{\mathrm{op}}.^[57] This construction yields an abelian group that classifies central simple algebras up to Morita equivalence.^[56] Alternatively, \mathrm{Br}(k) is isomorphic to the second Galois cohomology group H^2(\mathrm{Gal}(\bar{k}/k), \bar{k}^\times), where \bar{k} is a fixed algebraic closure of k.^[57] Key properties of the Brauer group include its torsion nature, as every element has finite order.^[58] For an element [\alpha] \in \mathrm{Br}(k) represented by a central simple algebra A of dimension d^2, the period of [\alpha] is the order of [\alpha] in \mathrm{Br}(k), while the index is the degree over k of a maximal separable subfield of a division algebra in the class (equivalently, the square root of the dimension of the representing division algebra).^[58] The period always divides the index, and the period-index problem investigates the precise relationship between these invariants, with known bounds such as index dividing period squared in characteristic zero.^[58] Representative examples illustrate the structure of Brauer groups. Over the rationals \mathbb{Q}, quaternion algebras provide 2-torsion elements; for instance, the Hamilton quaternion algebra (\frac{-1,-1}{\mathbb{Q}}), generated by i, j with i^2 = j^2 = -1 and ij = -ji, is a nonsplit division algebra of dimension 4, representing a nontrivial element of order 2 in \mathrm{Br}(\mathbb{Q}).^[59] More generally, quaternion algebras (\frac{a,b}{k}) over a field k of characteristic not 2 classify the 2-torsion subgroup of \mathrm{Br}(k).^[59] Cyclic algebras offer higher-order examples: given a cyclic Galois extension L/k of degree n with Galois group generated by \sigma and a \in k^\times, the cyclic algebra (L/k, \sigma, a) is the k-vector space \bigoplus_{i=0}^{n-1} u^i L with multiplication rules u^n = a and u x = x^\sigma u for x \in L, which has dimension n^2 and represents an element of order dividing n in \mathrm{Br}(k).^[37] The Brauer group connects to Azumaya algebras, which generalize central simple algebras to those over commutative rings and are locally isomorphic to matrix algebras over the ring.^[56] For a field k, Azumaya k-algebras coincide with central simple k-algebras, and \mathrm{Br}(k) classifies them up to similarity.^[57] A field extension L/k is a splitting field for a central simple algebra A if A \otimes_k L \cong M_d(L) for some d, and the minimal degree of such an extension equals the index of [A] in \mathrm{Br}(k).^[56]

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Ore, O. (1933) Theory of Non-Commutative Polynomials. Annals of Mathematics, 34, 480-508. https://doi.org/10.2307/1968173. has been cited by the ...
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Section 59.61 (03R1): Brauer groups—The Stacks project
The Brauer group of K is the set \text{Br} (K) of similarity classes of finite central simple algebras over K, endowed with the group law induced by tensor ...
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[PDF] Brauer Groups - Kiran S. Kedlaya
A ring A is called a k-algebra if it contains a field k in its center and is a finite dimensional k-vectorspace. If A is a subalgebra of a k-algebra E, then.
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[PDF] Lecture 16: Cohomological Description of the Brauer Group
Lemma 16.9: The period of an element in the Brauer group divides its index. In particular, the period is always finite, and Br is torsion.
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[PDF] Quaternion algebras - John Voight
Mar 20, 2025 · They capture essential features of noncommutative ring theory, number theory, 𝐾- theory, group theory, geometric topology, Lie theory, functions ...