Let $k$ be a field of chracteristic $p>0$. I want to show that $M=k[t]/(t^p) \otimes_{k[x,y]/(x^p,y^p)} k[t]/(t^p)$ is not isomorphic to $k[t]/(t^p)$ as $k$-algebra. Here the map from $k[x,y]/(x^p,y^p) \to k[t]/(t^p) $ is $x \to t$ and $y \to t$. The other map $k[x,y]/(x^p,y^p) \to k[t]/(t^p) $ is $x \to t$ and $y \to 0$. So, the algebra structure of $k[t]/(t^p)$ is different on both sides.
is it true that $M$ is not isomorphic to $k[t]/(t^p)$ (as a $k$-algrebra )?