In abstract algebra, an element a of a ring R is called a left zero divisor if there exists a nonzero x in R such that ax = 0,[1] or equivalently if the map from R to R that sends x to ax is not injective.[a] Similarly, an element a of a ring is called a right zero divisor if there exists a nonzero y in R such that ya = 0. This is a partial case of divisibility in rings. An element that is a left or a right zero divisor is simply called a zero divisor.[2] An element a that is both a left and a right zero divisor is called a two-sided zero divisor (the nonzero x such that ax = 0 may be different from the nonzero y such that ya = 0). If the ring is commutative, then the left and right zero divisors are the same.

An element of a ring that is not a left zero divisor (respectively, not a right zero divisor) is called left regular or left cancellable (respectively, right regular or right cancellable). An element of a ring that is left and right cancellable, and is hence not a zero divisor, is called regular or cancellable,[3] or a non-zero-divisor. A zero divisor that is nonzero is called a nonzero zero divisor or a nontrivial zero divisor. A non-zero ring with no nontrivial zero divisors is called a domain.


One-sided zero-divisor



Zero as a zero divisor

There is no need for a separate convention for the case a = 0, because the definition applies also in this case:

Some references include or exclude 0 as a zero divisor in all rings by convention, but they then suffer from having to introduce exceptions in statements such as the following:

Zero divisor on a module

Let R be a commutative ring, let M be an R-module, and let a be an element of R. One says that a is M-regular if the "multiplication by a" map is injective, and that a is a zero divisor on M otherwise.[4] The set of M-regular elements is a multiplicative set in R.[4]

Specializing the definitions of "M-regular" and "zero divisor on M" to the case M = R recovers the definitions of "regular" and "zero divisor" given earlier in this article.

See also


  1. ^ Since the map is not injective, we have ax = ay, in which x differs from y, and thus a(xy) = 0.


  1. ^ N. Bourbaki (1989), Algebra I, Chapters 1–3, Springer-Verlag, p. 98
  2. ^ Charles Lanski (2005), Concepts in Abstract Algebra, American Mathematical Soc., p. 342
  3. ^ Nicolas Bourbaki (1998). Algebra I. Springer Science+Business Media. p. 15.
  4. ^ a b Hideyuki Matsumura (1980), Commutative algebra, 2nd edition, The Benjamin/Cummings Publishing Company, Inc., p. 12

Further reading