Thread: Abstract Algebra: a problem about ideals

Let J be the set of all polynomials with zero constant term in Z[x].
a.) Show that J is the principal ideal (x) in Z[x].
b.) Show that Z[x]/J consists of an infinite number of distinct cosets, one for each nZ.

Originally Posted by iwonde

Let J be the set of all polynomials with zero constant term in Z[x].
a.) Show that J is the principal ideal (x) in Z[x].
b.) Show that Z[x]/J consists of an infinite number of distinct cosets, one for each nZ.

I can't see the whole question because there is a "physicsforums.com" tag that hides some of your question.

(a) is simply to verify that J is an ideal in Z[x] (Note that Z[x] is not a P.I.D. )
(b) Z[x] / (x) \cong Z. You can find a surgective group homomorphism from Z[x] to Z whose kernel is (x). Then by the first isomorphism theorem, Z[x] / (x) \cong Z. It means Z[x]/J has an infinite number of distinct cosets.