Given a module $M$ over a ring $R$, is it possible to endow $M$ with an operation $M^{2} \to M$ that turns $M$ to an algebra that is not the trivial $m n \equiv \mathbf{0}$ identically? So far I have that if $M$ is a free module, i.e. it has a basis, then we can define a pointwise multiplication that non-trivially makes $M$ an algebra.
My first thought after this was to consider $M$ as a direct sum $R \oplus N$ for some submodule $N \subseteq M$, and define it by $(r, n)(r', n') = (r r', 0)$, but even this (if my module can be written as the direct sum as earlier supposed) feels kinda trivial. So the next question: can any module $M$ be endowed with multiplication such that $N$ is a nontrivial subalgebra for every nontrivial submodule $N \subseteq M$?
I'm quite at a loss on how to show it. Thanks.