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 $M/S$ withscalar 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 forsome $s\in S$ we have $x^{\prime }=x+s$ and therefore

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

$=\lambda x+\lambda s$

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 definesthe quotient vector space of a vector space by a vector subspace.