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# 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.

Defines:

quotient vector space

Type of Math Object:

Definition

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## Mathematics Subject Classification

16D10*no label found*

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