Let $R$ be a commutative ring with $1_R$.
I found in this post the definition of the action of a group acting on a ring. And then, the following questions came in my mind.
Question 1. What is the formal and common definition of a ring acting on a group? (Without use of equivalent definition of action with endomorphisms, as wiki does)
So, from this to conclude that "A module is a ring action on an abelian group".
Question 2. Now, let's remind to ourselves the following definition.
Definition. An abelian group $(A,+)$ is an associative $R$-algebra if: (1) $(A,+,\cdot) $ is a ring (not necessarily commutative), (2) $(A,+)$ together with the scalar multiplication $*:R\times A \longrightarrow A,\ (r,a) \longmapsto r*a$ is a left $R-$module and (3) the operations $*$ and $\cdot$ are compatible, that is $$\forall a,b \in A, \forall r\in R: r*(a\cdot b)=(r*a)\cdot b=a\cdot (r*b).$$
In this answer, we have a good way to think about the last axiom, as a result of an action.
Does the ring $R$ acts on the abelian group $(A,+)$ through the map (scalar multiplication) $R\times A \longrightarrow A,\ (r, a)\longmapsto r*a$ ? And if the answer is yes, why? In other words, what are precisely the properties which should are satisfied, in order to have this kind of action?
I'm trying to connect $R$-algebras with Group Theory.
Thank you very much.
The common way to think about the ring R acting on a group G is for there to be a formal multiplication $$R \times G \rightarrow G$$ $$(r, g) \mapsto r\cdot g$$,
that moves group elements around. This multiplication agrees with the group action in the sense that it's distributive.
In practice, the action is clear--matrices on vectors, multiplication of numbers, etc. In general, the ring action has to be told.