I am trying to understand the Kunneth formula in Hatcher's Algebraic Topology. Theorem 3.15 says
The cross product $H^∗(X;R)⊗_RH^∗(Y;R)→H^∗(X×Y;R)$ is an isomorphism of rings if $X$ and $Y$ are CW complexes and $H^k(Y;R)$ is a ﬁnitely generated free $R$-module for all $k$.
I want to apply this theorem on computing the cohomology ring of the torus. But from the definition tensor product of rings I get $\Bbb Z[x]/\langle x^2\rangle \otimes \Bbb Z[y]/\langle y^2\rangle=\Bbb Z[x,y]/\langle x^2,y^2\rangle$ which is not the standard result $\mathbb Z[x,y]/\langle x^2,y^2,xy+yx\rangle$.
So where does the relation $xy+yx$ comes from? How do we calculate the tensor product of graded rings here?