Modules over a ring

Abbreviation: RMod

Definition

A \emph{module over a rings with identity} RR is a structure A=A,+,,0,fr (rR)A=A,+,,0,fr (rR) such that

A,+,,0A,+,,0 is an abelian groups

frfr preserves addition: fr(x+y)=fr(x)+fr(y)fr(x+y)=fr(x)+fr(y)

f1f1 is the identity map: f1(x)=xf1(x)=x

fr+s(x))=fr(x)+fs(x)fr+s(x))=fr(x)+fs(x)

frs(x)=fr(fs(x))frs(x)=fr(fs(x))

Remark: frfr is called \emph{scalar multiplication by rr}, and fr(x)fr(x) is usually written simply as rxrx.

Morphisms

Let AA and BB be modules over a ring RR. A morphism from AA to BB is a function h:ABh:AB that is a group homomorphism and preserves all frfr:

h(fr(x))=fr(h(x))h(fr(x))=fr(h(x))

Examples

Example 1:

Basic results

Properties

Finite members

f(1)=1f(2)=f(3)=f(4)=f(5)=f(6)=

Subclasses

Superclasses

References


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