How can I calculate $\mathbb Z[x]/x^2 \times \mathbb Z[y]/y^2 \times \mathbb Z[z]/z^2$?

Question

I am guessing that $\mathbb Z[x]/x^2 \times \mathbb Z[y]/y^2 \times \mathbb Z[z]/z^2 = \mathbb Z[x,y,z]/(x^2, y^2, z^2, xy, yz, xz)$ but I do not have an educated justification for this if it is true. If it is wrong, could someone explain to me the correct solution?

Answer 1

The suggested isomorphism is wrong. The ring $$\mathbb Z[x,y,z]/(x^2, y^2, z^2, xy, yz, xz)$$ has only two idempotents, while the other ring has eight idempotents.

If $f(f-1)\in (x^2, y^2, z^2, xy, yz, xz)$, then $f(0)(f(0)-1)=0$. Suppose $f(0)=0$. It follows that $f\in(x,y,z)$. Then $f^2\in (x^2, y^2, z^2, xy, yz, xz)$, so $f\in(x^2, y^2, z^2, xy, yz, xz)$.