Module

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Important for the Rings and Modules. Demote when fleshed out

Definition

Let [ilmath](R,+,*,0)[/ilmath] be a ring - not necessarily with unity - then a "left [ilmath]R[/ilmath]-module" is:

• An Abelian group, [ilmath](M,\oplus)[/ilmath] together with a
• left action, [ilmath][:R\times M\rightarrow M][/ilmath] given by [ilmath][:(r,x)\mapsto rx][/ilmath] of [ilmath]R[/ilmath] on [ilmath]MM[/ilmath], called the "left [ilmath]R[/ilmath]-module structure" on [ilmath]M[/ilmath]

such that:

1. [ilmath]\forall r,s\in R,\forall x\in M[r(sx)=(rs)x][/ilmath],
2. [ilmath]\forall r,s\in R,\forall x\in M[(r+s)x=rx+sx][/ilmath] and
3. [ilmath]\forall r\in R,\forall x,y\in M[r(x+y)=rx+ry][/ilmath]

Additionally, if [ilmath]R[/ilmath] is a u-ring then a left [ilmath]R[/ilmath]-module is unital when:

4. [ilmath]\forall x\in M[1_Rx=x][/ilmath]

The notation [ilmath]{}_RM[/ilmath] generally indicates that [ilmath]M[/ilmath] is a left [ilmath]R[/ilmath]-module

See next

• Direct product of modules - an instance of a product
• External direct sum of modules - an instance of a co-product
• Homomorphism