modules over algebars and homomorphisms between them

Let R be a ring and let A be an associative algebra (not necessarily unital).

Definition. A (left) A-module over R is a pair (M,) where M is a (left) R-module and


is a R-bilinear map such that the following conditions hold:

  1. 1.


  2. 2.


for any a,bA, xM and rR. We will simply use capital letters to denote modules.

Let M be an A-module over R. If MM and AA then by AM we denote R-submoduleMathworldPlanetmath of M generated by elements of the form am for aA and mM. We will call M unitary if AM=M. Note, that if A has multiplicative identityPlanetmathPlanetmath 1, then M is unitary if and only if 1m=m for any mM.

The reason we use name ,,A-module over R” instead of ,,A-module” is that these to concepts may differ. The latter means that we treat A simply as a ring and take modules over it. But such module need not be equiped with a ,,good” R-module structureMathworldPlanetmath. On the other hand this is always the case, when M is unitary over unital algebraPlanetmathPlanetmath.

If M and N are two A-modules over R, then a function f:MN is called an A-homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath iff f is an R-homomorphism and additionaly f(am)=af(m) for any aA and mM.

It can be easily checked that A-modules over R together with A-homomorphisms form a categoryMathworldPlanetmath which is abelianMathworldPlanetmath. Furthermore, if A is unital, then its full subcategory consisting unitary R-modules over A is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to category of unitary A-modules.

In most cases it is important to assume that the base ring R is a field, even algebraically closedMathworldPlanetmath.

Title modules over algebars and homomorphisms between them
Canonical name ModulesOverAlgebarsAndHomomorphismsBetweenThem
Date of creation 2013-03-22 19:16:32
Last modified on 2013-03-22 19:16:32
Owner joking (16130)
Last modified by joking (16130)
Numerical id 5
Author joking (16130)
Entry type Definition
Classification msc 13B99
Classification msc 20C99
Classification msc 16S99