Idea to make tensor product of two module a module structure

Question

If M is an (S,R) bimodule and N is an left R module then How to make tensor product of M and N a S module.While making it an S module structure i got stuck in proving the property of scalar product distributive over addition.How to do that?

Answer 1

The typical element of $M\otimes_T N$ has the form $\sum_{i=1}^k m_i\otimes n_i$. Then $S$ acts on the left on $M\otimes_R N$ by $$s\sum_{i=1}^k m_i\otimes n_i=\sum_{i=1}^k (sm_i)\otimes n_i.$$ To be more precise, the map $\phi_s:M\times N\to M\otimes_R N$ defined by $$\phi_s(m,n)=(sm)\otimes n$$ is bilinear and $R$-balanced, so induces a map $\psi_s:M\otimes_RN \to M\otimes_RN$. One needs to check that $\psi_1$ is the identity, $\psi_{s'}+\psi_s=\psi_{s'+s}$ and $\psi_{s'}\circ\psi_s=\psi_{s's}$. It suffices to do that on elements of the form $m\otimes n$.