I try to re-ask this question in a most precise way since it was closed.
Let $k$ be a field, and let $R_{a,b}=k[x,y,z]/ (x^az,xy^b)$, for some positive integers $a,b$. I have to find a primary decomposition of $(0)$ and for which $a,b$ there are embedded primes.
Now since a decomposition is $(x),(x^a,y^b),(z,y^b)$, the associated primes are the respective radicals, i.e. $(x),(x,y),(z,y)$ (for any $a,b$). It is for this reason that I don't understand why the existence of embedded primes depends on $a,b$, and is not true that $(x,y)$ is always embedded, containing $(x)$.
$I$ is a monomial ideal, and we can find a primary decomposition in a straightforward way: $$(x^az,xy^b)=(x^a,xy^b)\cap(z,xy^b)=(x^a,x)\cap(x^a,y^b)\cap(z,x)\cap(z,y^b)=(x)\cap(x^a,y^b)\cap(z,y^b).$$ In particular, this shows that $\dim R_{a,b}=2$.