Module over a commutative unital ring

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This article is about a basic definition in commutative algebra. View a complete list of basic definitions in commutative algebra


Let R be a commutative unital ring. A module over R is an Abelian group M along with a map .: R \times M \to M such that:

a.(b.m) = (ab).m \ \forall \ a,b \in R, \ m \in M


1.m = m \ \forall \ m \in M

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

(a + b).m = a.m + b.m \ \forall \ a,b \in R, \ m \in M

It follows that 0.m = 0 and (-a).m = -(a.m)

  • The map m \mapsto a.m is an endomorphism of M, viewed as an Abelian group.

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