Finding prime ideals of $\mathbb{Z}[x,y]/(12,x^2,y^3)$

Question

I am trying to find the prime ideals of $\mathbb{Z}[x,y]/(12,x^2,y^3)$, how could one go about doing this? Is there any general strategy to follow?

My ideas are that by the fourth isomorphism theorem, any ideal in $\mathbb{Z}[x,y]/(12,x^2,y^3)$ must be an ideal in $\mathbb{Z}[x,y]$ that contains $(12,x^2,y^3)$. But I don't know how to use this piece of information to find the prime ideals in the quotient ring. I also know that any ideal $I$ in $\mathbb{Z}[x,y]/(12,x^2,y^3)$ will be prime iff $\mathbb{Z}[x,y]/(12,x^2,y^3)/I$ is an integral domain. But again, I'm not sure how knowing this helps me find the prime ideals of the quotient ring. Thanks for your help!

Answer 1

You need that a prime ideal is generated by prime factors, each generated by a prime factor of the generators of $(12,x^2,y^3)$. This is $$(2,x,y), (3,x,y).$$ The main idea is: Think as you annihilate the things that produce zero-divisors.

Answer 2

My take is that the above quotient ring is ${\Bbb Z}_{12}[x,y]/\langle x^2,y^3\rangle$, where each element is of the form $a+b\bar x + c\bar y + d\bar y^2 + e\bar x\bar y$ with coefficients in ${\Bbb Z}_{12}$. Now expand Jyrki's comment.