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.