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Zero divisor

In abstract algebra, particularly within the theory of rings, a zero divisor is a nonzero element a in a ring R such that there exists another nonzero element b \in R with a \cdot b = [0](/page/0) or b \cdot a = [0](/page/0), where \cdot denotes the ring's multiplication operation.^[1]^[2] More precisely, if the multiplication is non-commutative, one distinguishes between left zero divisors (where a \cdot b = [0](/page/0) for some nonzero b) and right zero divisors (where b \cdot a = [0](/page/0) for some nonzero b); in commutative rings, these coincide.^[2] The zero element itself is trivially a zero divisor in any nontrivial ring, but the concept focuses on nonzero instances, which prevent the ring from satisfying the zero product property—wherein the product of two nonzero elements is nonzero.^[1] Commutative rings with multiplicative identity lacking nonzero zero divisors are termed integral domains, a fundamental class that includes the integers \mathbb{Z} and polynomial rings over fields, enabling key theorems like unique factorization.^[2]^[3] Classic examples abound in quotient rings: in \mathbb{Z}/6\mathbb{Z}, the elements 2 and 3 are zero divisors since $2 \cdot 3 = 0 \pmod{6}, yet neither is zero.^[2] Similarly, in the ring of $2 \times 2 matrices over the reals, the matrices \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} and \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} multiply to the zero matrix.^[1] Zero divisors play a crucial role in commutative algebra, influencing ideal structure, localization, and the study of modules, as their presence signals deviations from domain-like behavior essential for algebraic geometry and number theory.^[2]

Definition and Fundamentals

Definition

In ring theory, a ring is an algebraic structure consisting of a set R equipped with two binary operations, addition and multiplication, such that (R, +) forms an abelian group, multiplication is associative, and the distributive laws hold: a(b + c) = ab + ac and (a + b)c = ac + bc for all a, b, c \in R.^[4] A nonzero element a \in R is called a left zero divisor if there exists a nonzero element b \in R such that a b = [0](/page/0), where [0](/page/0) denotes the additive identity of the ring.^[5] Similarly, a nonzero element a \in R is a right zero divisor if there exists a nonzero element b \in R such that b a = 0.^[5] An element that is both a left zero divisor and a right zero divisor is called a two-sided zero divisor.^[5] In a commutative ring, where multiplication satisfies a b = b a for all a, b \in R, the notions of left, right, and two-sided zero divisors coincide./02%3A_Fields_and_Rings/2.02%3A_Rings) Rings without nonzero zero divisors are known as integral domains.^[3]

Relation to Units and Integral Domains

In ring theory, units are elements that possess multiplicative inverses, meaning for an element u in a ring R, there exists u^{-1} \in R such that u u^{-1} = u^{-1} u = 1, where $1 is the multiplicative identity. A zero divisor cannot be a unit, as the existence of a nonzero element b such that a b = 0 leads to a contradiction if a is assumed invertible: multiplying both sides by a^{-1} yields b = a^{-1} (a b) = a^{-1} 0 = 0.^[6] This distinction underscores how zero divisors disrupt the invertibility essential to units, preventing rings with zero divisors from exhibiting field-like behavior for all nonzero elements.^[7] Integral domains represent a class of rings free from such disruptions. Specifically, an integral domain is defined as a commutative ring with unity (where the unity $1 \neq 0) that contains no zero divisors other than zero itself, ensuring that the product of any two nonzero elements is nonzero./16:_Rings/16.04:_Integral_Domains_and_Fields) Equivalently, a commutative ring with unity is an integral domain if and only if it has no zero divisors, as this condition directly enforces the absence of nontrivial zero products.^[8] This characterization classifies rings based on the presence or absence of zero divisors, with integral domains serving as a foundational structure analogous to the integers \mathbb{Z}, where divisibility behaves predictably without zero-induced collapses. Fields extend this structure further, forming integral domains in which every nonzero element is a unit. In a field, the lack of zero divisors combined with universal invertibility for nonzero elements ensures full division capability, distinguishing fields as the maximal integral domains under this criterion./16:_Rings/16.04:_Integral_Domains_and_Fields) Thus, while all fields are integral domains, the converse holds only when invertibility permeates the entire nonzero ring, highlighting the hierarchical relationship between zero divisors, units, and domain classifications.^[9]

Examples and Non-Examples

Concrete Examples

A concrete illustration of zero divisors appears in the ring of 2×2 matrices over \mathbb{Z}/6\mathbb{Z}, denoted M_2(\mathbb{Z}/6\mathbb{Z}). Consider the matrices A = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} and B = \begin{pmatrix} 3 & 0 \\ 0 & 2 \end{pmatrix}. Their product is AB = \begin{pmatrix} 6 & 0 \\ 0 & 6 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}, the zero matrix in this ring, while both A and B are nonzero.^[10] In the polynomial ring (\mathbb{Z}/6\mathbb{Z}), constant polynomials and nonconstant ones can serve as zero divisors when the coefficient ring has them. For instance, the constant polynomial $3 (viewed as $3 + 0x + \cdots) and the linear polynomial $2x satisfy $3 \cdot 2x = 6x = 0 \cdot x = 0, with both nonzero in the ring. This follows from the general criterion that a polynomial is a zero divisor if the greatest common divisor of its coefficients and $6 exceeds $1.^[11] Direct products of rings with zero divisors also exhibit them prominently. In \mathbb{Z}/6\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z} under componentwise addition and multiplication, the elements (2, [0](/page/0)) and (3, [0](/page/0)) are nonzero, but their product is (2 \cdot 3, [0](/page/0) \cdot [0](/page/0)) = (6, [0](/page/0)) = ([0](/page/0), [0](/page/0)), the zero element. This pattern generalizes: in a direct product R \times S, elements like (r, [0](/page/0)) and (0, s) with r, s nonzero yield zero when multiplied, provided R and S are nonzero rings.^[12] Rings of functions from a set to a ring, with pointwise operations, provide another intuitive class of examples. Consider the ring of all functions f: X \to \mathbb{Z}/6\mathbb{Z} where X has at least two elements, say X = \{1, 2\}, under pointwise addition and multiplication. Let \chi_1 be the characteristic function with \chi_1(1) = 1 and \chi_1(2) = 0, and \chi_2 with \chi_2(1) = 0 and \chi_2(2) = 1. Then \chi_1 \cdot \chi_2 is the zero function, as it evaluates to $0 at both points, while both \chi_1 and \chi_2 are nonzero. More generally, functions with disjoint supports multiply to the zero function in such rings.^[10]

Non-Examples in Common Rings

The ring of integers \mathbb{Z} provides a fundamental non-example of a ring containing zero divisors. In \mathbb{Z}, the product of two nonzero elements is always nonzero, as \mathbb{Z} is an integral domain where ab = 0 implies a = 0 or b = 0. This property follows directly from Euclid's lemma, which states that if a prime p divides the product ab, then p divides a or p divides b, ensuring no nontrivial zero divisors exist.^[13] Polynomial rings over fields similarly lack zero divisors. For a field k, the ring k of polynomials in one indeterminate x is an integral domain, meaning the product of two nonzero polynomials is nonzero. This absence of zero divisors is guaranteed by Gauss's lemma, which asserts that the content of a product of primitive polynomials is the product of their contents, leading to unique factorization and precluding zero divisors in such rings.^[14] Fields themselves, such as the rational numbers \mathbb{Q} or the real numbers \mathbb{R}, contain no zero divisors by definition. In a field, every nonzero element has a multiplicative inverse, so if ab = 0 with a \neq 0, then multiplying both sides by a^{-1} yields b = 0. These structures—\mathbb{Z}, k for fields k, and fields—are all integral domains, serving as essential building blocks in abstract algebra for constructing quotient rings, modules, and other extensions without the disruptions caused by zero divisors.^[9]^[15]

Algebraic Properties

Basic Properties

In a ring R, let Z(R) denote the set of zero divisors. The set Z(R) is not necessarily closed under addition, as there exist rings where the sum of two zero divisors is neither zero nor a zero divisor; for example, in the direct product ring \mathbb{R} \times \mathbb{R}, the elements (1,0) and (0,1) are zero divisors, but their sum (1,1) is a unit.^[16] However, Z(R) exhibits closure properties under multiplication: the product of a non-zero element and a zero divisor is either zero or itself a zero divisor. Specifically, if a \in Z(R) with a \neq 0 and there exists b \neq 0 such that ab = 0, then for any r \in R with r \neq 0, (ra)b = r(ab) = 0, so ra annihilates b \neq 0; if additionally ra \neq 0, then ra \in Z(R).^[17] In a commutative ring R, if a \in Z(R) is a zero divisor, then any non-zero multiple ka for k \in R \setminus \{0\} with ka \neq 0 is also a zero divisor. To see this, suppose ab = 0 for some b \neq 0; then (ka)b = k(ab) = 0, so ka annihilates the non-zero element b. This property highlights the "propagation" of zero-divisor behavior under scaling by non-zero elements in commutative settings.^[11] The presence of zero divisors in a ring prevents the validity of cancellation laws for multiplication. In particular, if R has a zero divisor a \neq 0, there exist b, c \in R with b \neq c but ab = ac; for instance, if ab = 0 = ac with b \neq c and both non-zero. Conversely, a ring satisfies the cancellation laws—if ab = ac and a \neq 0 implies b = c—if and only if it has no zero divisors. This equivalence underscores the role of zero divisors in obstructing unique factorization and division-like operations.^[18] For a non-zero element a \in R, the (left) annihilator is defined as \operatorname{Ann}(a) = \{x \in R \mid ax = 0\}. If a is a zero divisor, then \operatorname{Ann}(a) \neq \{0\}, since there exists some non-zero b with ab = 0, so b \in \operatorname{Ann}(a). In commutative rings, the annihilator is two-sided, and zero divisors are precisely the non-zero elements with non-trivial annihilators. This concept links zero divisors to ideal theory, as \operatorname{Ann}(a) is always an ideal.^[16] Regarding ideals, the set of left zero divisors in a general ring does not always form a left ideal, as it may fail closure under addition: if a and a' are left zero divisors with annihilators generated by distinct non-zero elements whose product vanishes, the sum a + a' may not annihilate any non-zero element. However, in commutative Noetherian rings, the set of zero divisors coincides with the union of the associated prime ideals.^[16]^[19]

Properties in Specific Ring Structures

In quotient rings, the presence of zero divisors is closely tied to the nature of the ideal. Consider a commutative ring R with an ideal I. The quotient ring R/I consists of cosets \bar{r} = r + I for r \in R, with multiplication defined by \bar{a} \bar{b} = \overline{ab}. Thus, \bar{a} \bar{b} = \bar{0} if and only if ab \in I.^[20] Zero divisors in R/I arise from non-zero cosets \bar{a}, \bar{b} (i.e., a \notin I, b \notin I) such that ab \in I; these may lift from zero divisors in R (where ab = 0 \subseteq I) or from elements whose products lie in I without either being in I, such as when one annihilates a non-zero element of I.^[20] A key theorem states that R/I is an integral domain (hence has no zero divisors) if and only if I is a prime ideal; therefore, if I is not prime, R/I contains zero divisors, assuming R/I \neq 0.^[20] For direct products of rings, zero divisors emerge prominently even when the component rings lack them. Given commutative rings R and S with unity, the direct product R \times S has componentwise addition and multiplication, so elements are pairs (r, s) with r \in R, s \in S. If both R and S are non-trivial (i.e., not the zero ring), then (1_R, 0_S) and (0_R, 1_S) are non-zero elements satisfying (1_R, 0_S) \cdot (0_R, 1_S) = (0_R, 0_S), making them zero divisors.^[12] More generally, R \times S always contains zero divisors unless at least one of R or S is the zero ring, as the construction inherently produces such pairs regardless of whether R or S individually has zero divisors.^[12] In local rings, the structure of the maximal ideal highlights potential zero divisors among non-units. A commutative ring A is local if it has a unique maximal ideal m, which comprises all non-units of A (i.e., elements outside m are units).^[21] While local rings may be integral domains (with no zero divisors), in general, the maximal ideal m can contain zero divisors; for instance, if A is not a domain, elements in m often annihilate non-zero elements within A, placing them among the non-units.^[21] This containment underscores how zero divisors, when present, reside in the non-unit ideal m, distinguishing local rings from fields (where m = \{0\}).^[21]

Special Cases

The Zero Element as a Zero Divisor

In any ring R, the zero element $0 satisfies the equation $0 \cdot b = 0 for every element b \in R, including all non-zero b. Thus, by a literal interpretation of the condition for a zero divisor—without the non-zero stipulation—$0 qualifies as both a left and right zero divisor, since there exist non-zero elements b such that $0 \cdot b = 0 = b \cdot 0./02:_Fields_and_Rings/2.02:_Rings) Standard definitions in ring theory, however, explicitly exclude the zero element by requiring a zero divisor to be a non-zero element a \in R such that there exists a non-zero b \in R with a \cdot b = [0](/page/0) or b \cdot a = [0](/page/0). This convention originated in early abstract treatments of rings, such as Abraham Fraenkel's 1914 work on zero divisors and ring decompositions, and was reinforced in Emmy Noether's 1921 paper on ideal theory in ring domains, where the focus on non-zero elements facilitated the study of factorization and ideal structures in non-commutative settings.^[22] Including the zero element as a zero divisor would trivialize the concept, as every ring (except the zero ring) possesses $0 with this property, eliminating its utility in classifying rings like integral domains, which are defined precisely as commutative rings with unity having no zero divisors.^[23] The zero element remains the unique absorbing element under multiplication in any ring, but it is not considered a proper zero divisor under this exclusionary convention.^[24]

One-Sided Zero Divisors

In non-commutative rings, zero divisors exhibit asymmetry, distinguishing left and right variants from the two-sided case prevalent in commutative settings. A non-zero element a \in [R](/page/R) of a ring R is a left zero divisor if there exists a non-zero b \in [R](/page/R) such that ab = 0; it is a right zero divisor if there exists a non-zero b \in [R](/page/R) such that ba = 0. An element satisfying both conditions is a two-sided zero divisor. These definitions extend the standard notion of zero divisors, capturing the directional nature of multiplication in non-commutative structures. In commutative rings, left and right zero divisors coincide due to the symmetry ab = ba, making all zero divisors two-sided by default. However, non-commutativity allows rings where an element is a left zero divisor without being a right zero divisor (or vice versa), illustrating how the lack of commutativity introduces one-sided behavior. This distinction is absent in commutative rings but fundamental to understanding more general algebraic structures, such as matrix rings over fields, where zero divisors often arise from rank deficiencies in products. A concrete example occurs in the \mathbb{Z}_2-algebra R generated by indeterminates x_i (i \in \mathbb{N}) subject to the relations x_i x_j = 0 for all i < j. Here, x_1 serves as a left zero divisor since x_1 x_2 = 0 with x_2 \neq [0](/page/0). Yet, x_1 is not a right zero divisor, as the right annihilator of x_1 contains only the zero element—no non-zero b \in R satisfies b x_1 = 0. This demonstrates a ring with asymmetric zero divisors, where left-sided annihilation is possible without a corresponding right-sided counterpart. In rings with unity, the existence of a one-sided zero divisor does not generally imply it is two-sided. For instance, the example above is a ring with unity (adjoining 1 if necessary preserves the relations), yet x_1 remains strictly left-sided. However, in specific classes of rings with unity, such as eversible rings—where every left zero divisor is also a right zero divisor and conversely—one-sided zero divisors effectively become two-sided. This property holds in reversible rings (a subclass of eversible rings) but fails in general non-commutative settings, emphasizing the need for additional structural assumptions.

Generalizations and Extensions

Zero Divisors in Modules

In module theory, the concept of zero divisors extends from rings to the action of a ring on a module. Let R be a ring and M an R-module. An element r \in R is called a zero divisor on M if there exists a nonzero element m \in M such that r m = 0.^[25] This generalizes the notion of zero divisors in rings, where the special case M = R recovers the standard ring-theoretic definition.^[26] The set of all elements in R that annihilate the entire module M forms the annihilator ideal \operatorname{Ann}_R(M) = \{ r \in R \mid r M = 0 \}, which is an ideal of R consisting precisely of those zero divisors on M that act trivially on every element of M.^[25] A module M is faithful if \operatorname{Ann}_R(M) = 0, meaning no nonzero element of R acts as a zero divisor on the whole M, so the natural map R \to \operatorname{End}_R(M) is injective.^[26] In contrast, torsion modules highlight the pervasive role of zero divisors: an R-module M is torsion if every nonzero m \in M has a nonzero annihilator \operatorname{Ann}_R(m) = \{ r \in R \mid r m = 0 \} \neq 0, i.e., some nonzero r \in R acts as a zero divisor on m.^[26] For submodules, the zero divisors behave inclusion-wise. If N is a submodule of M, then the set of zero divisors on N is a subset of those on M: any r \in R that annihilates a nonzero element of N also annihilates a nonzero element of M.^[25] More precisely, the annihilator satisfies \operatorname{Ann}_R(N) \subseteq \operatorname{Ann}_R(M), since N \subseteq M implies that elements killing N kill a subset of M.^[25] This containment reflects the hierarchical structure of module actions under zero divisors.

Zero Divisors in Non-Commutative Settings

In non-commutative rings, the concept of zero divisors extends beyond commutative settings, where left and right zero divisors may differ, but the focus here is on their implications for ring structure and advanced algebraic properties. Division rings, or skew fields, are fundamental examples of non-commutative rings with no nonzero zero divisors: every nonzero element admits a left and right inverse, generalizing the structure of fields while allowing non-commutative multiplication. This absence of zero divisors ensures that the ring is a domain, and every finite division ring is in fact commutative, hence a field, by Wedderburn's little theorem.^[27] A key characterization in Artinian rings states that any Artinian ring with no zero divisors is a division ring, reflecting the rigid structure imposed by the descending chain condition on ideals. This result follows from the fact that such rings are semisimple Artinian, decomposing into matrix rings over division rings, but the lack of zero divisors forces the decomposition to a single division ring component. More generally, non-commutative domains (rings without zero divisors) satisfying the Ore condition embed into their classical ring of right quotients, which is a division ring, extending the framework to non-Artinian cases.^[28] In non-commutative algebras, orders—full lattice subrings of finite index in the algebra—often inherit or exhibit zero divisor properties from the ambient structure. For instance, the non-commutative polynomial ring (or free algebra) over an integral domain, such as the free algebra k\langle x_1, \dots, x_n \rangle in non-commuting indeterminates over a field k, is itself a domain with no zero divisors, as products of nonzero polynomials are nonzero due to the free generation and the base domain's integrity. However, in more general non-commutative polynomial rings over rings with zero divisors, such as matrix rings, zero divisors propagate, leading to complex factorization behaviors distinct from commutative cases. Unlike commutative rings, where the presence of zero divisors implies the ring is not prime (as the zero ideal would not be prime), non-commutative zero divisors can occur in prime rings without violating primeness. A prime non-commutative ring has no two nonzero two-sided ideals I, J such that IJ = 0, yet it may contain zero divisors if they do not generate such annihilating ideal pairs; examples include certain simple Artinian rings or enveloping algebras with specific zero divisor configurations that preserve ideal products. This distinction highlights how non-commutativity allows for richer ideal structures where zero divisors do not immediately force non-primeness. Modern applications leverage zero divisors to detect structural degeneracy in non-commutative settings. In Lie algebras, absolute zero divisors—nonzero elements x such that [x, [x, L]] = 0—signal degeneracy, as non-degenerate Lie algebras have no such nonzero elements, ensuring the adjoint representation is faithful in the sense of no nonzero nilpotent elements of index 2; simple Lie algebras over algebraically closed fields of characteristic zero, for example, lack such divisors unless exceptional.^[29] Similarly, in operator algebras, positive zero divisors in C*-algebras serve as tools to construct hereditary C*-subalgebras and invariants, detecting non-trivial ideals or non-simplicity that indicate degeneracy in the representation theory or K-theoretic properties of the algebra.^[30]

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