Are $\mathbb{C}[x]/x^2 \otimes \mathbb{C}[x]/x^2$ and $\mathbb{C}[x]/x^2$ isomorphic as $\mathbb{C}[x]/x^2$-modules?

Question

Are $\frac{\mathbb{C}[x]}{x^2} \otimes \frac{\mathbb{C}[x]}{x^2}$ and $\frac{\mathbb{C}[x]}{x^2}$ isomorphic as $\frac{\mathbb{C}[x]}{x^2}$-modules? I believe that they are but it I know that there is a mistake in my working and this is the claim that I'm the most uncertain of.

Answer 1

For any ring $R$ and any $R$-module $M$ we have $M\otimes_RR\cong M$ canonically. Taking $R=M=\Bbb{C}[x]/(x^2)$ yields the desired result.