# Random Ring theory Questions

1. Apr 13, 2007

I know that Z/(p) that is the integers mod a prime ideal is a field
and I also know that:
Field -> Euclidean Domain -> Principal Ideal domain -> Unique factorization domain ->Domain
So I know that Z/(p) are all of these things.
I also know that Z/(a) That is the set of integers mod a non prime number is not a field. But is it any of the other things? That is it a Euclidean Domain, Principal Ideal domain, Unique factorization domain,Domain, or noetherian?
I have the same question about R[x] and R[x,y] By this notation I mean the sets of polinomials with real coefficients in one and two variable. I believe that R[x] has all of the above listed properties. But I was wondering about R[x,y] I beilve it is not a field but it is noetherian, but I dont know about the other properties.....

Can any one clear these things up? I am just trying to come to a more general understanding of what properties are the same and different in different rings. Thanks

Last edited: Apr 13, 2007
2. Apr 13, 2007

### AKG

Z/(a) is not a domain when a is composite, and this is very easy to prove, so do it. Since these rings are finite, they are Noetherian because there are only finitely many subsets of these rings, hence only finitely many ideals (so any increasing chain of ideals must certainly stabilize).

R[x] is not a field, what would the inverse of x be? It is a Euclidean domain though. The Euclidean algorithm for polynomial division is something you should have learned in high school, it's just long division. Both R[x] and R[x,y] are Noetherian according to Wikipedia.