Which are nilpotent elements of $\mathbb{Q}[x]/(x^5-3x^2)\times\mathbb{Z}/(12)$?
I tried to decompose in this way: $$\mathbb{Q}[x]/(x^5-3x^2)\times\mathbb{Z}/(12)\cong\mathbb{Q}[x]/(x^2)\times\mathbb{Q}[x]/(x^3-3)\times\mathbb{Z}/(3)\times\mathbb{Z}/(4)$$ so i thought that nilpotent elements are only: $$(0,0,0,2), (x,0,0,2) \ \ \mbox{and} \ \ (x,0,0,0).$$
I don't know if I am right, because i tried another approach considering the intersetion of all prime ideals of that ring and i don't know to understand if the result is the same.
The nilpotent elements of the product are obtained as the tuples of the nilpotent elements of single factors.