Module

A module over a ring , in the left-module convention, is an Abelian group together with an operation , written , satisfying

			(1)
			(2)
			(3)

for all and (Dummit and Foote 2004, p. 337). If has a multiplicative identity 1, the additional condition defines a unital R-module. A module is similar to a vector space, but its coefficients lie in a ring rather than necessarily a field. A module over is also called an R-module.

Modules are the basic tool of homological algebra. Examples over the ring of integers include itself, the integer lattice , and the integral group ring of a group. Each uses ordinary integer multiplication as its coefficient operation.

is a module over itself. It is closed under addition and subtraction (although it is sufficient to require closure under subtraction). Numbers of the form for and a fixed integer form the submodule since, for all ,

(4)

and is still in .

Given two integers and , the smallest submodule of containing them is , where is their greatest common divisor.