examples of modules

This entry is a of examples of modules over rings. Unless otherwise specified in the example, M will be a module over a ring R.

  • Any abelian groupMathworldPlanetmath is a module over the ring of integersMathworldPlanetmath, with action defined by ng for gG given by ng=i=1ng.

  • If R is a subring of a ring S, then S is an R-module, with action given by multiplicationPlanetmathPlanetmath in S.

  • If R is any ring, then any (left) ideal I of R is a (left) R-module, with action given by the multiplication in R.

  • Let R= and let E={2kk}. Then E is a module over the ring of integers. Further, define the sets B=E×E and C=E×{0} and D={0}×E. Then B, C, and D are modules over ×, with action given by ax=(ax1,ax2) if x=(x1,x2) even if the product is redefined as ax1=0 and ax2=0, but now the identity elementMathworldPlanetmath is (1,1). However by our new product definition ax=(ax1,ax2)=(0,0) even if a=(1,1), the ring identity element originally In the more general definition of module which does not require an identity element 𝟏 in the ring and does not require 𝟏m=m for all mM, we observe that 𝟏mm in this example just constructed. (one of the purposes of this comment is to show that all modules need not be unital ones).

  • Yetter-Drinfel’d module. (http://planetmath.org/QuantumDouble)

Title examples of modules
Canonical name ExamplesOfModules
Date of creation 2013-03-22 14:36:28
Last modified on 2013-03-22 14:36:28
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 16
Author mathcam (2727)
Entry type Example
Classification msc 16-00
Classification msc 20-00
Classification msc 13-00
Related topic QuantumDouble
Related topic Module