Let $F$ be an algebraically closed field. Consider the polynomial ring $F[X,Y,Z]$ and its quotient by the ideal $J=(X^aZ, XY^b)$. I want to find all the $(a,b)$, for $a,b$ positive integers, such that there are no embedded primes in the primary decomposition of zero.
Unfortunately, I don't know how to use the fact that F is alg. closed to answer the question.
I computed that, for any a, b, the primary decomposition of $J$ will be the intersection of $(X)$, $(X^a,Y^b)$, $(Z,Y^b)$ (do you agree?), and that the minimal primes are $(X)$ and $(Z,Y)$. I nowhere used the hypothesis of $F$ alg. closed. The embedded prime shouldn’t be $(X,Y)$? I'm definitely missing something here...