Nonsimple Tensors (10.4.20)

Dummit and Foote Abstract Algebra, section 10.4, exercise 20:

Let $R=\mathbb{Z}[x]$ and let $I=(2,x)$. Show that the element $2 \otimes 2 + x \otimes x$ in $I \otimes_R I$ is not a simple tensor.

Proof: Define a map$$I \times I → R$$$$(i,j) \mapsto ij$$This is clearly seen to be $R$-balanced, and so induces a homomorphism $\Phi$ on $I \otimes_R I$. Assuming $2 \otimes 2 + x \otimes x = a \otimes b$, then also$$\Phi(2 \otimes 2 + x \otimes x)=\Phi(2 \otimes 2) + \Phi(x \otimes x)=x^2+4=\Phi(a \otimes b)=ab$$Since $x^2+4$ is a monomial quadratic with no roots, it does not factor in $R$, and thus either $a$ or $b$ is $\pm 1$ and now $a \otimes b∉I \otimes I$.$~\square$