This question is from Abstract Algebra book by Pierre Grillet (2nd edition).
Let $\phi:R\to S$ be a homomorphism of rings with identity and let $A$ be a unital left $S$-module. Make $A$ a unital left $R$-module.
Aim is to find a map $g:R \times A \to A$ that satisfies module conditions.
We have $\phi(1_R)=1_S$ and since $A$ is a unital left $S$-module, we have a function $f: S \times A \to A$ with $s \in S, a\in A, f(s,a)=sa$, and since it is unital, we also have $f(1_S,a)=a $ for all $a\in A$.
Can I define the function $g$ as $g:f\circ\phi$, the composition?
If it is possible, we have $(1_R,a)\to (1_S,a)\to a$, which gives unitarity.
Also commutativity is done, since for $r_1,r_2\in R, a\in A$, $R$ action is like, $(r_1+r_2)a=\phi(r_1+r_2)a=[\phi(r_1)+\phi(r_2)]a=\phi(r_1)a+\phi(r_2)a$ which is equal to $r_1a+r_2a$
But $r_1(r_2a)=(r_1r_2)a$ doesn't seem to hold because $\phi$ is a ring homomorphism and we can't have $\phi(r1r2)=r_1\phi(r_2)$.
I might be totally wrong on my work, if so, can someone help me find the function?