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Zero-divisors

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Given an integer in some ring , if there is some integer in that same ring such that , then is a zero-divisor. If , then is a trivial zero-divisor and might or might not be a zero-divisor. But if and either, yet , then both and are nontrivial zero-divisors. In some texts, "zero-divisor" means "nontrivial zero-divisor."

Most people take it for granted that 0 is the only zero-divisor in rings such as .[1] Consider the ring , where addition and multiplication are performed modulo 6:[2] for example, . Thus, in , 2, 3 and 4 are nontrivial zero-divisors since . It follows that if is composite, then has at least one nontrivial zero-divisor; in fact, there are nontrivial zero-divisors, where is Euler's totient function.

  1. John J. Watkins, Topics in Commutative Ring Theory. Princeton and Oxford: Princeton University Press (2007): 24
  2. See modular arithmetic.