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2008-05-12T16:27:12Z

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Vipul at 20:29, 5 January 2008

2008-01-05T20:29:43Z

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{{basicdef}}

==Definition==

Let <math>R</math> be a [[commutative unital ring]]. A '''module''' over <math>R</math> is an [[Abelian group]] <math>M</math> along with a map <math>.: R \times M \to M</math> such that:

* <math>.</math> is a monoid action of the [[multiplicative monoid]] of <math>R</math> on <math>M</math>, viz.:

<math>a.(b.m) = (ab).m \ \forall \ a,b \in R, \ m \in M</math>

and:

<math>1.m = m \ \forall \ m \in M</math>

* <math>.</math> is an additive homomorphism from <math>R</math> (treated as an additive group) to the additive group of all functions from <math>M</math> to itself, under pointwise addition. In symbols:

<math>(a + b).m = a.m + b.m \ \forall \ a,b \in R, \ m \in M</math>

It follows that <math>0.m = 0</math> and <math>(-a).m = -(a.m)</math>

* The map <math>m \mapsto a.m</math> is an endomorphism of <math>M</math>, viewed as an Abelian group.

All the above three conditions can be stated concisely as: the map <math>R \times M \to M</math> [[homomorphism of unital rings]] <math>f:R \to End(M)</math>, where <math>r.m := f(r)(m)</math>.

Module over a commutative unital ring - Revision history

Vipul: 1 revision

Vipul at 20:29, 5 January 2008