If $G$ is a group and $X$ is a set (possibly with some extra structure defined on it), then a group action of $G$ on $X$ is a map $a:G\times X\to X$ such that
- $a(1,x)=x$
- $a(g,a(h,x))=a(gh,x)$
Now, a vector space is often described as a field acting on an abelian group. The "acting" part of this description seems to refer to the following four axioms:
- $\lambda(u+v)=\lambda u+\lambda v$
- $(\lambda + \mu)v=\lambda v+\mu v$
- $1v=v$
- $(\lambda \mu )v=\lambda(\mu v)$
My question is twofold: is there a precise notion of a "field action" on some algebraic structure, and if so, how does it relate to group actions?
Of course, axioms (3) and (4) of a vector space bear a lot of resemblance to the axioms of a group action. Axiom (3) states that the identity element behaves in the way you expect it to, and axiom (4) is a kind of "associativity". But I wonder if the connection between these two types of "action" runs deeper than this.
An action of a monoid $M$ on a set $X$ is a monoid homomorphism $M\to\mathrm{End}(X)$. This makes sense because $$\mathrm{End}(X)=\{\text{functions } f:X\to X\}$$ is a monoid. If $M$ is a group, then an action of $M$ on $X$ is a group action.
Similarly, an action of a ring $R$ on an Abelian group $A$ is a ring homomorphism $R\to\mathrm{End}(A)$, where now $$\mathrm{End}(A)=\{\text{group homomorphisms } f:A\to A\}$$ is a ring. If $R$ is a field, then an action of $R$ on $A$ is a field action.