R-module

Noun

Meaning

R-module (plural R-modules)

(algebra) The monoid action of a ring R on an abelian group.

Examples

An element in a commutative ring R may be thought of as an endomorphism of any R-module. Source: Internet

For instance, if U is right R-module, and V is a maximal submodule of U, U·J(R) is contained in V, where U·J(R) denotes all products of elements of J(R) (the "scalars") with elements in U, on the right. Source: Internet

Conversely, suppose that M is a simple R-module. Source: Internet

Every simple R-module is isomorphic to a quotient R/m where m is a maximal right ideal of R. Herstein, Non-commutative Ring Theory, Lemma 1.1.3 By the above paragraph, any quotient R/m is a simple module. Source: Internet

If R is a ring and M is a right R-module, then the tensor product with M yields a functor F : R-Mod → Ab. Source: Internet

The latter is a generalization of the R-module example, since a ring can be understood as a preadditive category with a single object. Source: Internet

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