quotient module

Let M be a module over a ring R, and let S be a submodule of M. The quotient module M/S is the quotient groupMathworldPlanetmath M/S with scalar multiplication defined by λ(x+S)=λx+S for all λR and all xM.

This is a well defined operation. Indeed, if x+S=x+S then for some sS we have x=x+s and therefore

λx =λ(x+s)

so that λx+S=λx+λs+S=λx+S, since λsS.

In the special case that R is a field this construction defines the quotient vector space of a vector spaceMathworldPlanetmath by a vector subspace.

Title quotient module
Canonical name QuotientModule
Date of creation 2013-03-22 14:01:18
Last modified on 2013-03-22 14:01:18
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 9
Author rspuzio (6075)
Entry type Definition
Classification msc 16D10
Defines quotient vector space