Extending monoids to a ring

This was a classical question in semigroup theory. I suggest looking at this paper http://www.numdam.org/article/SD_1969-1970__23_2_A12_0.pdf which is interested in finite semigroups. It gives a reference to a Russian paper that in some sense says it impossible to axiomatize such rings but I don't have access to the Russian paper to say what sense.

A key obstruction is that commuting idempotents in a ring have joins and relative complements in rings but not semigroups. Another problem is rings with no zero divisors are cancelative but not semigroups.

I am not sure how checkable this is, but one formal answer is that the forgetful functor from rings to monoids has a left adjoint: the monoid ring functor. Rings are the same thing as algebras over the monad resulting from this adjunction. So the set of ring structures on a monoid M is the same as the set of monoid maps $\mathbb Z[M]\to M$ that satisfy the axioms of algebra over a monad.