Subalgebra

In abstract algebra, a subalgebra of a k-algebra A, where k is a commutative ring (often a field), is a nonempty subset B \subseteq A that is both a subring of A (closed under addition and multiplication, containing the multiplicative identity if A is unital) and a k-submodule of A (closed under scalar multiplication by elements of k).^[1] This structure ensures B inherits the algebraic operations from A while forming its own k-algebra, making subalgebras fundamental building blocks for studying algebraic structures such as rings, fields, and modules.^[1] Subalgebras appear in diverse contexts, including associative algebras like matrix rings M_n(k), where diagonal matrices form a commutative subalgebra,^[1] and non-associative algebras like Lie algebras, where a Lie subalgebra is a subspace closed under the Lie bracket [x,y].^[2] For instance, the complex numbers \mathbb{C} form a 2-dimensional subalgebra of the quaternions \mathbb{H} over the reals \mathbb{R},^[1] and they facilitate homomorphisms and ideals in algebraic extensions. In universal algebra, the notion extends to algebraic systems beyond rings, where a subalgebra is any subset closed under all operations of the parent structure.^[3]

General Concepts

Definition

In algebra, an algebra A over a commutative ring R (or more specifically, over a field when R is a field) is defined as a ring with multiplicative identity $1_A that is also an R-module, equipped with a bilinear multiplication operation satisfying r(ab) = (ra)b = a(rb) for all r \in R and a, b \in A.^[1] When R is a field, A is a vector space over R with the multiplication being R-bilinear.^[4] This structure ensures that the scalar multiplication from R commutes with the ring operations in A.^[5] A subalgebra S of an algebra A over R is a subset S \subseteq A that is itself an R-algebra under the induced operations from A, meaning S is closed under addition, scalar multiplication by elements of R, and the multiplication in A, and contains the multiplicative identity $1_A of A.^[1] Equivalently, S is both an R-submodule of A and a subring of A with the same identity.^[4] This closure respects the full algebraic structure, including the compatibility between module actions and ring multiplication.^[5] While a subspace (or submodule) of A is merely closed under addition and scalar multiplication, a subalgebra additionally requires closure under the bilinear multiplication and inclusion of the identity, thereby inheriting the complete algebra operations rather than just the module structure.^[1] This distinction highlights that subalgebras preserve the ring-theoretic aspects essential to the algebra's definition.^[5]

Basic Properties

A subalgebra S of an algebra A over a commutative ring R is a nonempty subset that is closed under the addition and multiplication operations inherited from A, as well as under scalar multiplication by elements of R.^[1] This closure ensures that S forms both an R-submodule of A and a subring of A, thereby inheriting the full algebraic structure of A restricted to S.^[1] In the context of universal algebra, a subalgebra is similarly defined as a subset closed under all fundamental operations of the parent algebra, preserving the induced structure without additional assumptions on the ring R.^[6] If the parent algebra A is unital, possessing a multiplicative identity $1_A, then unital subalgebras are required to contain $1_A, ensuring they share the same unit element and form unital algebras in their own right.^[1] Non-unital subalgebras, by contrast, need not contain $1_A but still satisfy the closure properties, allowing for structures like ideals or proper subspaces that lack a global identity.^[7] This distinction is particularly relevant in associative settings, where unital subalgebras preserve invertibility properties tied to the identity. The subalgebra relation is transitive: if S is a subalgebra of T and T is a subalgebra of A, then S is necessarily a subalgebra of A, as the closure under operations in T implies closure in the larger structure of A. This transitivity follows directly from the definitions and underpins the lattice structure of subalgebras within an algebra.^[6] For any subset X \subseteq A, the subalgebra generated by X, denoted \langle X \rangle, is the smallest subalgebra containing X, constructed as the intersection of all subalgebras containing X.^[7] Explicitly, \langle X \rangle consists of all finite R-linear combinations of finite products of elements from X, providing the minimal structure closed under the algebra's operations that incorporates X.^[1] In particular, for a singleton \{a\}, the generated subalgebra R is commutative and spans the polynomials in a with coefficients in R.^[1]

Subalgebras in Ring and Field Algebras

Associative Algebras over Fields

In the context of associative algebras over a field k, a subalgebra B of an associative k-algebra A is defined as a subset of A that is both a k-subspace (closed under addition and scalar multiplication by elements of k) and a subring with respect to the multiplication in A (closed under multiplication and containing additive inverses).^[8] This refinement leverages the vector space structure inherent to algebras over fields, distinguishing subalgebras from those in more general ring settings where module properties may complicate closure. If A is unital, subalgebras are typically required to contain the identity element $1_A to preserve the unital structure.^[5] A prominent example of such a subalgebra arises in polynomial rings, where the subring k \subseteq k[x,y] consists of all polynomials in the single indeterminate x with coefficients in k, embedded within the polynomial ring k[x,y] in two indeterminates. This inclusion is closed under multiplication because the product of any two elements f(x), g(x) \in k is f(x)g(x), which depends only on x and yields no terms involving y, thus remaining in k. Similarly, addition and scalar multiplication preserve this form, confirming k as a subalgebra; moreover, it is infinite-dimensional over k with basis \{1, x, x^2, \dots \}.^[9] When A is finite-dimensional over k, any subalgebra B \subseteq A inherits this property and satisfies \dim_k B \leq \dim_k A, as B is a subspace of the finite-dimensional vector space A. A basis for B can be extended to a basis for A via the standard subspace dimension theorem, ensuring that the algebraic structure of B embeds compatibly within A without exceeding the ambient dimension.^[10] The center Z(A) = \{ a \in A \mid ab = ba \ \forall b \in A \} forms a subalgebra of A, as it is a k-subspace (closed under addition and scalars) and closed under multiplication: for a, c \in Z(A), (ac)b = a(cb) = a(bc) = (ac)b, confirming associativity and commutativity with all elements. The commutator subalgebra, often denoted [A, A] and spanned by elements of the form ab - ba for a, b \in A, also qualifies as a subalgebra in this setting, capturing the "non-commutative part" of A while respecting the associative multiplication.

Algebras over Rings

In the context of algebras over a commutative ring R, an R-algebra A is defined as an associative ring with identity that is also an R-module, equipped with a bilinear multiplication satisfying \alpha(ab) = (\alpha a)b = a(\alpha b) for all \alpha \in R and a, b \in A. A subalgebra B of A is then a subset that is both a subring (with the same identity) and an R-submodule of A, ensuring closure under the ring multiplication and the R-action. This definition generalizes the field case by replacing vector space structure with module structure, introducing challenges such as the lack of invertibility for scalars, which prevents automatic division and requires explicit verification of submodule closure.^[1] Unlike subrings, which may ignore the module structure, subalgebras must respect the R-action fully. For instance, consider the \mathbb{Z}-algebra \mathbb{Z}, where the structure map is the identity. The even integers $2\mathbb{Z} form a subring under addition and multiplication, but it is not a subalgebra because it fails to contain the multiplicative identity $1 \in \mathbb{Z}, and scalar multiplication by odd integers (viewed through the ring action) aligns with this exclusion, as subalgebras require sharing the unit element. In contrast, ideals like $2\mathbb{Z} are \mathbb{Z}-submodules but not subalgebras due to the unit condition. This highlights how the module and unital requirements distinguish subalgebras from mere subrings over rings like \mathbb{Z}.^[1] Bimodule aspects arise naturally since, for commutative R, the R-action on A is central, making A an R-bimodule with left and right actions coinciding. A subalgebra B \subseteq A is thus an R-bimodule closed under the internal multiplication of A. The centralizer of a subalgebra B, defined as C_A(B) = \{ a \in A \mid ab = ba \ \forall b \in B \}, forms a subalgebra containing B and the image of R, acting as the largest subalgebra commuting elementwise with B. Two-sided ideals of A are A-bimodules (and hence R-bimodules), but only the unit ideal is typically a subalgebra; non-unit ideals illustrate bimodule structure without unital closure. These centralizers and ideals play key roles in decomposition theorems for algebras over rings, such as analyzing simple components via centralizer chains.^[11] Regarding Noetherian properties, if R is a commutative Noetherian ring and A is a Noetherian R-algebra, then any subalgebra B that is finitely generated as an R-algebra inherits the Noetherian property, satisfying the ascending chain condition on ideals. This follows from the fact that finitely generated algebras over Noetherian base rings are themselves Noetherian, extending the Hilbert basis theorem to the module setting. However, arbitrary subalgebras need not inherit this property, as infinite generation can lead to non-stabilizing ideal chains, though specific classes (e.g., graded subalgebras in certain polynomial-like algebras) may preserve it under additional hypotheses.^[12]

Subalgebras in Universal Algebra

Universal Algebra Framework

In universal algebra, an algebra is defined as a nonempty set equipped with a collection of finitary operations of specified arities, collectively referred to as the type of the algebra, where these operations satisfy certain identities that define the algebraic structure.^[13] This framework abstracts common properties across diverse algebraic systems, such as groups, rings, and lattices, by focusing on operations and their preservation under structure-preserving maps rather than specific axioms.^[13] A subalgebra of an algebra A is a subset B \subseteq A that is closed under all the operations of A, meaning that for every operation f of arity n in the type, if b_1, \dots, b_n \in B, then f^B(b_1, \dots, b_n) = f^A(b_1, \dots, b_n) \in B.^[13] This closure ensures that B itself forms an algebra of the same type, inheriting the identities satisfied by A, and it constitutes a substructure that preserves the operational relations of the parent algebra.^[13] Subalgebras thus provide a natural way to study embedded structures within larger algebras, analogous to subgroups in group theory but generalized to arbitrary operation sets. Subalgebras frequently appear as homomorphic images in universal algebra. A homomorphism \alpha: A \to B between algebras of the same type is a function that preserves operations, satisfying \alpha(f^A(a_1, \dots, a_n)) = f^B(\alpha(a_1), \dots, \alpha(a_n)) for all operations f and elements a_i \in A.^[13] The image \alpha(A) under such a map is itself a subalgebra of B, as it is closed under the operations of B by the preservation property.^[13] Conversely, kernels of homomorphisms correspond to congruences on A, which are equivalence relations compatible with the operations, and the quotient algebra A / \ker(\alpha) is isomorphic to the image \alpha(A), linking subalgebras to these quotient constructions.^[13] Free subalgebras are generated by subsets within the context of a variety, which is a class of algebras closed under homomorphic images, subalgebras, and products, and defined by identities.^[13] For a subset X of an algebra A in a variety \mathcal{V}, the subalgebra generated by X, denoted \mathrm{Sg}^A(X), is the smallest subalgebra containing X, consisting of all elements obtainable by evaluating terms (polynomial expressions built from the operations and constants) applied to elements of X.^[13] This generated subalgebra is free in \mathcal{V} on X if X satisfies no additional relations beyond those enforced by the identities of \mathcal{V}, making it isomorphic to the free algebra in \mathcal{V} on X.^[13] Such free generations highlight the role of terms in constructing substructures without imposed dependencies.

Varietal Subalgebras

In universal algebra, a variety is a class of algebras of the same type defined by a set of identities, which are equations that hold universally for all elements and operations in the algebras, such as the associative law in groups or the distributive law in rings.^[14] These equational classes ensure that all algebras within the variety share the same structural properties enforced by the identities.^[14] Subalgebras within a variety are subsets closed under all operations of the ambient algebra and thus automatically satisfy the defining identities, as identities are preserved under restrictions to operation-closed subsets.^[14] Congruence relations, which are equivalence relations compatible with the operations via the substitution property, further characterize the structure: in a variety, congruences on an algebra induce quotient algebras that remain within the variety, and subalgebras inherit compatible congruence classes.^[14] For instance, in the variety of groups—defined by identities for associativity, identity element, and inverses—a subalgebra is a subgroup, which must contain the identity, be closed under the binary multiplication operation, and closed under the unary inverse operation to preserve the group structure.^[14] Normal subgroups serve as subalgebras in this variety while also generating principal congruences, as their cosets form the equivalence classes compatible with group operations, ensuring quotients are groups.^[14] Birkhoff's HSP theorem implies that varieties are closed under subalgebras (S), homomorphic images (H), and products (P), meaning the subalgebras of any algebra in the variety, along with their homomorphic images and arbitrary products, remain within the variety. This closure property underscores the robustness of varietal subalgebras, facilitating the study of algebraic structures through decomposition and projection.^[14]

Advanced Topics and Examples

Subalgebras in Lie Algebras

A Lie algebra over a field k (typically \mathbb{R} or \mathbb{C}) is defined as a vector space \mathfrak{g} equipped with a bilinear operation [\cdot, \cdot]: \mathfrak{g} \times \mathfrak{g} \to \mathfrak{g}, called the Lie bracket, that is skew-symmetric ([x, y] = -[y, x] for all x, y \in \mathfrak{g}) and satisfies the Jacobi identity ([x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 for all x, y, z \in \mathfrak{g}).^[15] This structure captures infinitesimal symmetries, contrasting with associative algebras by emphasizing non-associative bracket relations derived from commutators.^[15] A sub-Lie algebra (or subalgebra) of \mathfrak{g} is a subspace \mathfrak{h} \subseteq \mathfrak{g} that is closed under the Lie bracket, meaning [\mathfrak{h}, \mathfrak{h}] \subseteq \mathfrak{h}.^[15] Since the bracket is bilinear and the Jacobi identity holds globally, any such \mathfrak{h} automatically inherits the full Lie algebra structure, including skew-symmetry and the Jacobi identity.^[15] Subalgebras play a central role in decomposing \mathfrak{g} and studying its representations, often serving as building blocks for classifications like the Cartan decomposition. A prominent example arises in the special linear Lie algebra \mathfrak{sl}(n, \mathbb{C}), consisting of n \times n complex matrices with trace zero under the commutator bracket [A, B] = AB - BA. The Borel subalgebra \mathfrak{b} comprises all trace-zero upper triangular matrices, which forms a sub-Lie algebra because the product of two upper triangular matrices is upper triangular, and thus their commutator preserves the trace-zero condition and upper triangular form.^[15] Specifically, if A = (a_{ij}) and B = (b_{ij}) are trace-zero upper triangular (i.e., a_{ij} = b_{ij} = 0 for i > j), then [A, B]_{ij} = \sum_k (a_{ik} b_{kj} - b_{ik} a_{kj}) vanishes for i > j since the terms involve only upper triangular entries, ensuring closure.^[15] This \mathfrak{b} is maximal solvable and decomposes as \mathfrak{b} = \mathfrak{n} \oplus \mathfrak{h}, where \mathfrak{h} is the Cartan subalgebra of diagonal trace-zero matrices (abelian) and \mathfrak{n} is the nilpotent subalgebra of strictly upper triangular matrices.^[15] Within Lie subalgebras, solvability and nilpotency are characterized by descending series of nested subalgebras. A subalgebra \mathfrak{h} is solvable if its derived series terminates at zero: define \mathfrak{h}^{(0)} = \mathfrak{h} and \mathfrak{h}^{(k+1)} = [\mathfrak{h}^{(k)}, \mathfrak{h}^{(k)}] for k \geq 0, requiring \mathfrak{h}^{(m)} = \{0\} for some m.^[15] For the Borel subalgebra \mathfrak{b} \subset \mathfrak{sl}(n, \mathbb{C}), the first derived subalgebra \mathfrak{b}^{(1)} = \mathfrak{n} (strictly upper triangular), and subsequent terms shift superdiagonals until vanishing after n-1 steps, confirming solvability.^[15] Nilpotency strengthens this, using the lower central series: \mathfrak{h}_0 = \mathfrak{h} and \mathfrak{h}_{k+1} = [\mathfrak{h}, \mathfrak{h}_k], terminating at zero for some step. The nilradical \mathfrak{n} of \mathfrak{b} exemplifies this, as repeated bracketing with \mathfrak{b} (or itself) produces matrices with zeros on more initial superdiagonals, reaching zero in n-1 steps.^[15] These properties underpin root space decompositions and semisimple Lie algebra theory.^[15]

Relation to Ideals and Subrings

In the context of ring theory, a subring of a ring R is a subset closed under the ring's addition and multiplication operations, along with additive inverses, but when R is viewed as an algebra over a base ring or field k, a subalgebra requires additional closure under scalar multiplication by elements of k.^[16] This distinction arises because subrings need not respect the module structure over k, whereas subalgebras do. For instance, the integers \mathbb{Z} form a subring of the rationals \mathbb{Q}, as they are closed under integer addition and multiplication, but \mathbb{Z} is not a subalgebra of \mathbb{Q} over \mathbb{Q} itself, since scalar multiplication by \frac{1}{2} \in \mathbb{Q} maps $1 \in \mathbb{Z} to \frac{1}{2} \notin \mathbb{Z}.^[17] In associative algebras over a field, two-sided ideals play a special role as they are precisely the subalgebras that absorb multiplication from the ambient algebra on both sides. Specifically, a two-sided ideal I of an associative algebra A is closed under the algebra's multiplication and scalar actions, making it a subalgebra, while also satisfying A \cdot I \subseteq I and I \cdot A \subseteq I.^[5] This absorption property distinguishes ideals from general subalgebras, which may not interact with the full algebra in this way. However, simple associative algebras, which have no nontrivial two-sided ideals, can still contain proper subalgebras that are not ideals. Division algebras provide a key example: they are simple (with no proper two-sided ideals other than zero), yet they often admit proper subalgebras, such as maximal subfields embedded as commutative subalgebras. For instance, the real quaternions \mathbb{H} is a division algebra over \mathbb{R} with no proper ideals, but it contains \mathbb{C} as a proper subalgebra.^[18] The term "subalgebra" was formalized in the early 20th century amid the axiomatization of ring and algebra theory, particularly through works developing abstract structures beyond number fields and polynomials.^[19]