Let $A$ be a finite dimensional algebra over an algebraically closed field $K$, $M$ an $(A, K[t])$-bimodule that is free as a $K[t]$-module of rank $d$ and $N$ an irreducible left $K[t]$-module.
Why is the left $A$-module $M \otimes_{K[t]}N$ indecomposable?
I know that such $N$ is of the form $K[t]/(t-\lambda)$ for some $\lambda \in K$ and $M=\bigoplus_{i=1}^d m_i . K[t]$ for a basis $(m_1,\cdots,m_d)$, so that $M \otimes_{K[t]}N$ is $d$-dimensional as a vectorspace. Hence it suffices to show that $\mathrm{End}_{A}(M \otimes_{K[t]}N)$ is local.
I'm not getting any further than that and hope you can help me.