Recall that a commutative rng is a commutative ring, but possibly without identity.
Let $A$ be a commutative rng. The following proposition is a modified version of Nakayama's Lemma:
Proposition. For any finitely generated $A$-module $M$, there is an element $e\in A$ such that $em=m$ for all $m\in M$.
In a commutative rng $A$ any ideal $\mathfrak a\subset A$ is itself a subrng of $A$. If $M$ is finitely generated $A$-module, then by imposing the condition that $\mathfrak aM=M$, the $A$-module $M$ is then finitely generated over the rng $\mathfrak a$. In this case by applying the above proposition we have the familiar form of Nakayama's lemma.
My question arises from the above observation: Are there any other interesting examples of modules $M$ over a commutative rng $A$ without identity, but with the property below?
There is an element $e\in A$ such that $em=m$ for all $m\in M$.
One step further, my primary goal is to look for a larger class of $A$-modules for which the converse of the proposition could also hold.
Thank you.
Take any infinite collection of fields $\{F_i\mid i\in I\}$ and form the rng $R=\oplus_{i\in I} F_i$, by which I mean the elements of $\prod F_i$ of finite support with coordinatewise addition and multiplication. As is well known, $R$ lacks an identity.
Now, you can take any nontrivial idempotent $e\in R$ at all, and $M=eR$ is a module upon which $e$ acts like the identity. This should provide an ample number of examples.