Math 562 Spring 2012 Homework 4 Drew Armstrong
نویسنده
چکیده
Proof. First note that the zero element of the ring R/I is 0 + I = I and that a + I = I if and only if a ∈ I. Now suppose that I ⊆ R is a prime ideal and consider nonzero cosets a+ I and b+ I in R/I (i.e. consider a 6∈ I and b 6∈ I). Since I is prime this implies that ab 6∈ I, hence ab + I 6= I and we conclude that R/I is an integral domain. Conversely, let R/I be an integral domain and consider a, b ∈ R with ab ∈ I (i.e. consider ab+ I = I). Since (a+ I)(b+ I) = ab+ I = I and R/I is an integral domain we conclude that either a+ I = I (i.e. a ∈ I) or b+ I = I (i.e. b ∈ I). Hence I ⊆ R is a prime ideal. Now let I ⊆ R be a maximal ideal. You showed on the previous homework that this implies that R/I is a field. Since every field is an integral domain, we conclude that I is a prime ideal.
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