Prime ideals in commutative rings

Abstract

Prime ideals. One of the most important properties of a prime integer p is that a product of two integers is divisible by p only if at least one of the integers is divisible by p. In the notation of principal ideals, this can be restated as follows. If p is a prime, then n 1 n 2 ≡ 0(p) implies that n 1 ≡ 0(p) or n 2 ≡ 0(p) , or both. Now if R is an arbitrary commutative ring, an ideal p in R is said to be a prime ideal if and only if ab ≡ 0(p) implies that a ≡ 0(p) or b ≡ 0(p) , or both. Thus the prime ideals in I are precisely the ideals (p) , where p is a prime, together with the ideal (0) and the ideal I. In any commutative ring R, it is obvious that the ideal R is always prime; and the ideal (0) is a prime ideal if and only if R has no proper divisors of zero.