Zero divisor

In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero divisor if there exists a nonzero in such that . 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. An element  that is both a left and a right zero divisor is called a two-sided zero divisor (the nonzero such that may be different from the nonzero such that ). 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 is called left regular or left cancellable. Similarly, an element of a ring that is not a right zero divisor is called 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, or a non-zero-divisor. A zero divisor that is nonzero is called a nonzero zero divisor or a nontrivial zero divisor. A nonzero ring with no nontrivial zero divisors is called a domain.
Zero as a zero divisor.
There is no need for a separate convention for the case , because the definition applies also in this case: 
Some references include or exclude 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 be a commutative ring, let be an -module, and let be an element of . One says that is -regular if the "multiplication by " map formula_59 is injective, and that is a zero divisor on otherwise. The set of -regular elements is a multiplicative set in .
Specializing the definitions of "-regular" and "zero divisor on " to the case recovers the definitions of "regular" and "zero divisor" given earlier in this article.