zero divisor

In abstract algebra, an element  of a ring  is called a left zero divisor if there exists a nonzero 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 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.