Given a ring R and a left R-module T, a subset A of T is called a (left) submodule of T, if  (A,+)  is a subgroupMathworldPlanetmathPlanetmath of  (M,+)  and  raA  for all elements r of R and a of A.


  1. 1.

    The subsets {0} and T are always submodules of the module T.

  2. 2.

    The set  {tT:rt=trR}  of all invariant elements of T is a submodule of T.

  3. 3.

    If  XT  and 𝔞 is a left idealMathworldPlanetmathPlanetmath of R, then the set


    is a submodule of T.  Especially, RX is called the submodule generated by the subset X; then the elements of X are generatorsPlanetmathPlanetmath of this submodule.

There are some operationsMathworldPlanetmath on submodules.  Given the submodules A and B of T, the sumA+B:={a+bT:aAbB}  and the intersectionDlmfMathworldPlanetmath AB are submodules of T.

The notion of sum may be extended for any family  {Aj:jJ}  of submodules:  the sum jJAj of submodules consists of all finite sums jaj where every aj belongs to one Aj of those submodules.  The sum of submodules as well as the intersection jJAj are submodules of T.  The submodule RX is the intersection of all submodules containing the subset X.

If T is a ring and R is a subring of T, then T is an R-module; then one can consider the productPlanetmathPlanetmath and the quotient of the left R-submodules A and B of T:

  • AB:={finiteνaνbν:aνA,bνBν}

  • [A:B]:={tT:tBA}

Also these are left R-submodules of T.

Title submodule
Canonical name Submodule
Date of creation 2013-03-22 15:15:26
Last modified on 2013-03-22 15:15:26
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 19
Author PrimeFan (13766)
Entry type Definition
Classification msc 20-00
Classification msc 16-00
Classification msc 13-00
Related topic SumOfIdeals
Related topic QuotientOfIdeals
Defines R-submodule
Defines generated submodule
Defines generator
Defines sum of submodules
Defines product submodule
Defines quotient of submodules