Extension of scalars

Extension of scalars

Let ${\displaystyle {}_{S}S{}_{R}}$ denote ${\displaystyle S}$ regarded as an ${\displaystyle (S,R)}$-bimodule, where ${\displaystyle s_{1}\cdot s_{2}\cdot r=s_{1}\,s_{2}\,\rho (r)}$ for ${\displaystyle s_{1},s_{2}\in S}$ and ${\displaystyle r\in R}$. For a given left ${\displaystyle R}$-module ${\displaystyle V}$, the induced module ${\displaystyle \rho _{!}V}$ is defined as the tensor product of bimodules