Let me start with a construction. Take a commutative rng $A$ and for every finitely generated $A$-module $M$ take the endomorphism ring $\mathrm{End}_A(M)$ which is itself a ring with identity. Let the rng homomorphism $h_M: A\to\mathrm{End}_A(M)$ defined by mapping $a\in A$ to the map $m\mapsto am$. Now by the fact that $M$ is finitely generated over $A$ we must have some $a\in A$, $h_M(a)$ maps to the identity map in $\mathrm{End}_A(M)$.(To see this, apply Cayley-Hamilton for the identity map in $\mathrm{End}_A(M)$ to obtain an element $a\in A$ such that $am=m$ for all $m\in M$.) Thus the image $h_M(A)$ is a commutative ring with identity(It is commutative since it only contains left multiplications by elements in $A$.) Note that $h_M$ is injective if and only if $M$ is faithful. Thus
A commutative rng $A$ is a ring if and only if there exists a finitely generated faithful module acting on $A$.
There are quite a lot questions I came up with when trying to get some insight into this construction. So I guess the best answer should be a reference book/article on this subject, but let me write some of my questions down:
How canonical is $h_M$ for each finitely generated module $M$? Given another finitely generated module $N$ over $A$ is there a unique automorphism $f\in\mathrm{Aut}_A(M)$ such that $h_N=fh_M$?
How does informations for the map $h_M$ for each finitely generated module $M$ over $A$ tell us about the "obstruction" for adding an identity to the ring $A$? This question might be a little vague, but I should be looking for a "homological" answer.
What are the maps $h_M\in\mathrm{End}_A{M}$ for finitely generated module $M$ over $A$ tell us about the ring $A$?
Again the best answer I would wish to get should be a reference that includes this subject to study. Thank you in advance.