Let $A$, $B$ be two $k$-algebras of finite dimension, where $k$ is a field. Here, $A$ and $B$ are not necessarily commutative. Do we have that$$Z(A \otimes_k B) \cong Z(A) \otimes_k Z(B),$$where $Z(-)$ is the center?
2025-01-13 01:55:53.1736733353
Tensor product of $k$-algebras, center, isomorphism.
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