Direct calculations of $\text{ass}_R(R/I)$, $\text{supp}_R(R/I)$ and associate prime decomposition of ideal $I \subset R$

Question

Suppose we have $R = k[x, y, z]$ and product of two ideals $I = (x, y)(x^2, z) = (x^3, xz, yx^2, yz)$. I am trying to calculate directly $\text{ass}_R(R/I)$, $\text{supp}_R(R/I)$ and associate prime decomposition of $I$. I have proved that $I$ is not primary ideal, also i proved that $\text{ass}(M) \subset \text{supp}(M)$, but it seems to me that this gives me nothing. How to calculate this three things?

Answer 1

The support of $I$ is $\{(x,y,p(z))\mid p=0\text{ or irreducible}\} \cup \{(x,q(y),z)\mid q =0\text{ or irreducible}\}.$ Let us denote $S:=R/I.$ Then $\operatorname{Ann}(x^2)=(x,y,z), \operatorname{Ann}(yx)=(x,z), \operatorname{Ann}(z)=(x,y) $ ($x^2,yx,z$ are elements of $S$ and right hand sides are ideals of $R$) so that these three primes belong to $\operatorname{ass}_R(S)$. On the other hand,$I=(x^2,z)\cap(x^3,xz,yx^2,yz,z^3,y^3)\cap(x,y)$ and this is a shortest primary decomposition. Moreover, we have equality $\operatorname{ass}_R(S)=\{(x,y),(x,y,z),(x,z)\}$.