Let $R$ be a commutative ring with unity and $M$ an $R$-module. The mapping $$\begin{array}{ll}R&\to& \text{End}_{\Bbb Z}(M)\\a&\mapsto &f_a\,,\end{array}$$ which associates with $a\in R$ the homothety $$f_a(m)= ma\quad\forall m\in M\,,$$ is an isomorphism of $R$ into the ring $\text{End}_{\Bbb Z}(M)$ of endomorphisms of the additive group of $M$.
Questions.
(1) If $R$ is Artinian, is $A_M:=\{f_a\mid f_a:M\to M\}$ an Artinian ring.
(2) Does $A_M$ have a prime ideal?
The map you're suggesting isn't an isomorphism unless $M$ is faithful.
But anyway, $A_M$ is a homomorphic image of an Artinian ring, so it is also Artinian. Any ring with an identity (different from zero) has prime ideals, so of course this ring will too, as long as it is nonzero. In that case its identity is $f_1$.