# 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 group $M/S$ with scalar multiplication defined by $\lambda(x+S)=\lambda x+S$ for all $\lambda\in R$ and all $x\in M$.

This is a well defined operation. Indeed, if $x+S=x^{\prime}+S$ then for some $s\in S$ we have $x^{\prime}=x+s$ and therefore

 $\displaystyle\lambda x^{\prime}$ $\displaystyle=\lambda(x+s)$ $\displaystyle=\lambda x+\lambda s$

so that $\lambda x^{\prime}+S=\lambda x+\lambda s+S=\lambda x+S$, since $\lambda s\in S$.

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

Title quotient module QuotientModule 2013-03-22 14:01:18 2013-03-22 14:01:18 rspuzio (6075) rspuzio (6075) 9 rspuzio (6075) Definition msc 16D10 quotient vector space