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In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero such that ,〔See Bourbaki, p. 98.〕 or equivalently if the map from to that sends to is not injective.〔Since the map is not injective, we have = , in which differs from , and thus (-) = 0.〕 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.〔See Lanski (2005).〕 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 zero divisor is called regular, or a non-zero-divisor. A zero divisor that is nonzero is called a nonzero zero divisor or a nontrivial zero divisor. == Examples == * In the ring , the residue class is a zero divisor since . * The only zero divisor of the ring of integers is 0. * A nilpotent element of a nonzero ring is always a two-sided zero divisor. * A idempotent element of a ring is always a two-sided zero divisor, since . * Examples of zero divisors in the ring of matrices (over any nonzero ring) are shown here: *: *:. *A direct product of two or more nonzero rings always has nonzero zero divisors. For example, in ''R''1 × ''R''2 with each ''R''''i'' nonzero, (1,0)(0,1) = (0,0), so (1,0) is a zero divisor. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「zero divisor」の詳細全文を読む スポンサード リンク
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