Change of rings

In algebra, change of rings refers to one of several related ways for changing the coefficient ring of a module. Given (possibly non-commutative) unital rings ${\displaystyle R}$ and ${\displaystyle S}$ and a ring homorphism ${\displaystyle ho :R\to S}$,

• restriction of scalars turns an ${\displaystyle S}$-module ${\displaystyle V}$ into an ${\displaystyle R}$-module ${\displaystyle ho ^{*}V}$.
• extension of scalars turns an ${\displaystyle R}$-module ${\displaystyle V}$ into an ${\displaystyle S}$-module ${\displaystyle ho _{!}V}$, the induced module.
• coextension of scalars turns an ${\displaystyle R}$-module ${\displaystyle V}$ into an ${\displaystyle S}$-module ${\displaystyle ho _{*}V}$, the coinduced module.

In this article we describe the constructions on left modules, although all constructions work for right modules mutatis mutandis. Each of these constructions extends to a functor, so that we have an adjoint triple

where ${\displaystyle {}_{R}{\mathsf {Mod}}}$ and ${\displaystyle {}_{S}{\mathsf {Mod}}}$ are the categories of left ${\displaystyle R}$- and ${\displaystyle S}$-modules respectively.