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
Definition
Let be a commutative unital ring. A module over
is an Abelian group
along with a map
such that:
-
is a monoid action of the multiplicative monoid of
on
, viz.:
and:
-
is an additive homomorphism from
(treated as an additive group) to the additive group of all functions from
to itself, under pointwise addition. In symbols:
It follows that and
- The map
is an endomorphism of
, viewed as an Abelian group.
All the above three conditions can be stated concisely as: the map homomorphism of unital rings
, where
.