PlanetMath: module

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module

(Definition)

Let be a ring with identity. A left module over is a set with two binary operations, $+: M\times M \longrightarrow M$ and $\cdot: R \times M \longrightarrow M$ , such that

$(\u +\v )+\mathbf{w}= \u +(\v +\mathbf{w})$ for all $\u ,\v ,\mathbf{w}\in M$
$\u +\v =\v +\u$ for all $\u ,\v\in M$
There exists an element $\mathbf{0} \in M$ such that $\u +\mathbf{0}=\u$ for all $\u\in M$
For any $\u\in M$ , there exists an element $\v\in M$ such that $\u +\v =\mathbf{0}$
$a \cdot (b \cdot \u ) = (a \cdot b) \cdot \u$ for all $a,b \in R$ and $\u\in M$
$a \cdot (\u +\v ) = (a \cdot \u ) + (a \cdot \v )$ for all $a \in R$ and $\u ,\v\in M$
$(a + b) \cdot \u = (a \cdot \u ) + (b \cdot \u )$ for all $a,b \in R$ and $\u\in M$

A left module over is called unitary or unital if $1_R \cdot \u = \u$ for all $\u\in M$ .

A (unitary or unital) right module is defined analogously, except that the function $\cdot$ goes from $M \times R$ to and the scalar multiplication operations act on the right. If is commutative, there is an equivalence of categories between the category of left -modules and the category of right -modules.

"module" is owned by djao.

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See Also: maximal ideal, vector space

Other names:

left module, right module

Attachments:: modules are a generalization of vector spaces (Example) by jgade
examples of modules (Example) by mathcam
submodule (Definition) by Mravinci

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Cross-references: category, equivalence of categories, commutative, right, act on, operations, multiplication, scalar, function, unital, unitary, binary operations, identity, ring
There are 74 references to this entry.

This is version 6 of module, born on 2001-10-19, modified 2004-11-19.
Object id is 365, canonical name is Module.
Accessed 15707 times total.

Classification:

AMS MSC:	13-00 (Commutative rings and algebras :: General reference works )
	16-00 (Associative rings and algebras :: General reference works )
	20-00 (Group theory and generalizations :: General reference works )

Pending Errata and Addenda

None.[ View all 8 ]

Discussion

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simplifying the definition by hermann on 2008-04-19 11:40:08

Properties 1 through 4 are just the propreties of an abelian group. Wouldn't it be simpler to say that a module is an abelian group (M,+) with a binary operation . : R x M -> M that satisfies properties 5 through 7?

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typographical correction of unital module by remag12 on 2004-11-06 05:25:07

Left (right) R modules M for rings R with identity 1 such that 1 * m = m (m * 1 = m) for all m in M are called unital modules.
-- S. A. G.

[ reply | up ]

correction of typographicalerror of unital module by remag12 on 2004-11-06 05:17:56

Left (right) R modules M for rings R with identity 1 such that 1 * m = m (m * 1 = m) for all m in M are call unital modules.
-- S. A. G.

[ reply | up ]

correction of definition by remag12 on 2004-09-16 08:03:58

The module definition does not require an identity element in the ring. Those satisfying 1 * m = m for all m in the module are called
unitary modules.
-- S.A. G.

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