TBC: [[Group action]] on set $X$ corresponds to group of automorphisms ($\text{Sym}(X)$, bijective endomorphisms) which are a special case of [[Monoid action]] that correspond to monoid of endomorphisms under composition. Endomorphisms of [[Groups|abelian group]] inherit additive structure and form a ring with composition. Modules are a useful construction for studying abelian groups. In fact, any abelian group a can be made into a $\mathbb{Z}$-module which leads to the [[Fundamental Theorem of Finite Abelian Groups]]. # Definitions ###### Left $R$-module axioms Let $R$ be a ring with unity $1$. A left $R$-module $M$ consists of an abelian $(M,+_{M})$ and a 'scalar multiplication' operation $\cdot:R\times M\to M$ such that for all $r,s\in R$ and $x,y\in M$: (M1) Scalar multiplication distributes over module addition: $r\cdot(x+_{M}y)=r\cdot x+_{M} r \cdot y$ ; (M2) Compatibility of ring addition with module addition: $(r+_{R}s)\cdot x=r \cdot x+_{M}s \cdot x$ ; (M3) Associativity of scalar multiplication: $(rs)\cdot x=r\cdot(s \cdot x)$; (M4) Identity: $1\cdot x=x$. **Remarks (requiring $(M, +_M)$ to be abelian is not just a formality):** - The condition that $(M,+_{M})$ is abelian together with (M2) ensures that the ring addition (which, by definition, is commutative) is compatible with the module addition. - It follows from (M3) and (M4) that the unit group $R^{\times}$ acts on $M\setminus \{ 0 \}$ by scalar multiplication. The orbit space is the projectivization of the module. ###### In terms of 'ring action' TBC