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It's known that for a field $k$, the tensor product of $k$-vector spaces commutes with direct sums. Is it also true that the tensor product of $k$-algebras commutes with finite products ('finite products' in the ordinary sense of ring products)?

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The tensor product of vector spaces distributes over direct sums, so that $(V\oplus W)\otimes U\cong V\otimes U\oplus W\otimes U$ with a natural isomorphism. That isomorphism is an isomorphism of algebras if $U$, $V$ and $W$ are algebras.

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