Let $K$ be a field and let $\Lambda\_{1}$ and $\Lambda\_{2}$ be two finite-dimensional $K$-algebras with Jacobson radicals $J\_{1}$ and $J\_{2}$ respectively. How to show or where can I find the proof of the following statement?



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> $\Lambda\_{1} / J\_{1} \otimes\_{K} \Lambda\_{2} / J\_{2}$ is always semisimple if $K$ is perfect or if $\Lambda\_{1}$ and $\Lambda\_{2}$ are path algebras of quivers factored by admissible ideals.
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Thank you.



Tensor of finite-dimensional algebra over perfect field is semisimple
We have $gldim A \otimes\_K B= gldim A + gldim B$ if A and B are seperable algebras over the field $K$, see <https://www.cambridge.org/core/journals/nagoya-mathematical-journal/article/on-the-dimension-of-modules-and-algebras-viii-dimension-of-tensor-products/58116B52E52F0F6165E84AE11284CCF6> corollary 18.


Now being semisimple for finite dimensional algebras is equivalent to global dimension zero.


