Thread: Maximal Ideal

Let N = {f in Z[x] | f has constant term 0}, clearly N is an ideal of Z[x], show that N is a maximal ideal of Z[x].

Originally Posted by Coda202

Let N = {f in Z[x] | f has constant term 0}, clearly N is an ideal of Z[x], show that N is a maximal ideal of Z[x].

I don't beleive it is a maximal Ideal... Here is why

Define a Homomorphism from by

Note that the kernel of is N

is onto so we can use the first isomorphism theorem to get



Since is an Integral Domain this implies that The Ideal is Prime and not maximal.

This ideal in is contained in the Ideal gentered by (2, and the same set f) and this ideal is maximal