Module

Definition

$(M,R,\cdot )$

An algebraic structure consisting of a set $M$, an associated Ring $R$, and an operation of scalar multiplication $\cdot$ defined between the ring elements and the elements of the set where the scalar multiplication is distributive.

A generalization of the notation of Vector Space in which the field of scalars is replaced by a Ring

Let $r,s\in R$ and $x,y\in M$

• $r\cdot (x+y)=r\cdot x+r\cdot y$
• $(r+s)\cdot x=r\cdot x+s\cdot x$
• $(rs)\cdot x=r\cdot (s\cdot x)$
• $1\cdot x=x$