Module mathematics: Encyclopedia II - Module mathematics - Definition

Module mathematics - Definition

Specifically, a left module over the ring R consists of an abelian group (M, +) and an operation R × M → M (called scalar multiplication, usually just written by juxtaposition, i.e. as rx for r in R and x in M) such that

For all r,s in R, x,y in M, we have

1. r(x+y) = rx+ry
2. (r+s)x = rx+sx
3. (rs)x = r(sx)
4. 1x = x

Usually, we simply write "a left R-module M" or _RM. A right R-module M or M_R is defined similarly, only the ring acts on the right, i.e. we have a scalar multiplication of the form M × R → M, and the above axioms are written with scalars r and s on the right of x and y.

Authors who do not require rings to be unital omit condition 4 in the above definition, and call the above structures "unital left modules". In this article however, all rings and modules are assumed to be unital.

A bimodule is a module which is both a left module and a right module.

If R is commutative, then left R-modules are the same as right R-modules and are simply called R-modules.

Adapted from the Wikipedia article "Definition", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki