Module theory MOC

Module

A module over a ring is an abelian group together with an action of on that is distributive and -linear. module Explicitly, a left-module over satisfies the following for any and

whereas a right-module satisfies the same properties with scalar multiplication written on the right.¹ Thus a module is a generalization of a vector space, which is just a module over a field. This small change has far-reaching implications, for example the existence of Torsion.

Further terminology

• Finitely generated module

Properties

• A module inherits linear structure from the underlying ring

Examples

• Vector space
• Let be an ideal. Then is an -submodule of .
• Let be a ring extension. Then is an -module.

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Footnotes

1. If is a commutative ring the concepts of left- and right-modules coïncide, but otherwise there is a distinction between left- and right-scalar multiplication. ↩