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Definition: Let R be a ring. The following conditions on an ideal P of R are equivalent. An ideal that satisfies these conditions is called a prime ideal. a) The quotient ring R/P is an integral domain. b) If P is not equal to R, and if a and b are elements of R such that ab is in P, then a is in P or b is in P. c) If P is not equal to R, and if A and B are ideals of R such that AB is a subset of P, then A is a subset of P or B is a subset of P.
Source: Algebra, second edition by Michael Artin