For any $R$-module $M$, the module $S \otimes_R M$ is an $S$-module.

Question

In the book Commutative algebra written by N.S. Gopalakrishnan it has been stated as a definition that for any $R$-module $M$, the module $S \otimes_R M$ is an $S$-module for the operation defined by $$s(s_1 \otimes x)=ss_1 \otimes x,\ s_1 \in S,\ x \in M.$$ This $S$-module is called the module obtained by extension of scalars.

Now my question is how is that operation well-defined? Suppose $s=s'$ and $s_1 \otimes x = s_1' \otimes x'$ then how can I show that $ss_1 \otimes x = s's_1' \otimes x'$?

I have tried but couldn't able to figure out why? Please help me.

Thank you very much.

Answer 1

For each $s\in S$, prove that $(s_1,x)\mapsto (ss_1)\otimes x$ is an $R$-bilinear map from $S\times M$ to $S\otimes_R M$. Then it induces an $R$-linear map from $S\otimes_R M$ to $S\otimes_R M$.