PlanetMath: quotient module

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quotient module

(Definition)

Let be a module over a ring , and let be a submodule of . The quotient module is the quotient group 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 then for some $s\in S$ we have and therefore

$\displaystyle \lambda x'$	$\displaystyle = \lambda(x+s)$
	$\displaystyle = \lambda x+\lambda s$

so that $\lambda x' + S = \lambda x + \lambda s + S = \lambda x + S$ , since $\lambda s \in S$ .

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

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"quotient module" is owned by rspuzio. [ full author list (3) | owner history (2) ]

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Also defines:

quotient vector space

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Cross-references: vector subspace, vector space, field, operation, well defined, multiplication, scalar, quotient group, submodule, ring, module
There are 9 references to this entry.

This is version 6 of quotient module, born on 2003-10-15, modified 2006-07-29.
Object id is 4966, canonical name is QuotientModule.
Accessed 3453 times total.

Classification:

AMS MSC:

16D10 (Associative rings and algebras :: Modules, bimodules and ideals :: General module theory)

Pending Errata and Addenda

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