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Modules over a ring
Abbreviation: RMod
Definition
A \emph{module over a rings with identity} RR is a structure A=⟨A,+,−,0,fr (r∈R)⟩A=⟨A,+,−,0,fr (r∈R)⟩ such that
⟨A,+,−,0⟩⟨A,+,−,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)
fr∘s(x)=fr(fs(x))fr∘s(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:A→Bh:A→B 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)=