In his book "Algebra: Chapter 0", Paolo Aluffi gave the following exercise:
which prompts the following question: under which circumstance(s) does the the tensor product change with the change of ring ?
2026-04-03 10:35:53.1775212553
Differences in tensor products under change of ring
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I believe that the tensor product does not change precisely when the morphism of rings $R\to S$ is an epimorphism. The reason is that the tensor product over $S$ is constructed by balancing the tensor product over $R$ with respect to the $S$-action. But if the morphism is an epimorphism, the $S$-action is uniquely determined by the $R$-action and the tensor product over $R$ must therefore already be $S$-balanced.