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Let $A$ and $B$ be two commutative rings with unit. Let $\rho: A \rightarrow B$ be a ring homomorphism and $N$ a $B$-module (and thereby a $A$-module).

Question: Is there a canonical morphism $B\otimes_A N \to N$? If so, what is the expression of this morphism?

Lelong Wang
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Consider the $A$-bilinear map: $$\begin{align*} B\times N\to N \\ b\times n \to bn. \end{align*}$$ By the universal property, this induces a unique $A$-module (surjective) homomorphism $$B\otimes_{A} N\to N.$$

belkacem abderrahmane
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  • Your inverse map is not correct unless the tensor product is with respect to $R$, which is not the case in the question (also as a general rule try not to change the notations used in the question). – Captain Lama Nov 04 '22 at 07:19
  • I didn't pay attention that the tensor product being over $A$, I will edit my answer. – belkacem abderrahmane Nov 04 '22 at 07:23