# Zero divisor

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 elementÂ *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 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**,^{[3]} 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.

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:

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.