Module

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Important for the Rings and Modules. Demote when fleshed out

Definition

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

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

such that:

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

Additionally, if $R$ is a u-ring then a left $R$-module is unital when:

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

The notation ${}_RM$ generally indicates that $M$ is a left $R$-module

See next

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