# Extending monoids to a ring

I started reading about monoids (and semigroups in general) and came across of the example of some non-commutative monoids which cannot be endowed with some addition turning it into a ring (the monoid is constructed such that $$x^2=x$$ and so the ring would be boolean and thus commutative). My question is:

Do we have (checkable) criteria to decide whether a monoid can be turned into a ring?

• One obvious condition is that there must be an element $\mathbf 0$ satisfying $\mathbf 0\cdot x=x\cdot \mathbf 0=\mathbf 0$ for all $x$ Jan 4, 2021 at 9:25
• If $p$ is prime the only possible ring structures are zero (semigroup with 0 and constant law $xy=0$) and a cyclic group of order $p-1$ plus an absorbing element. This discards most monoid structures on a set with $p$ elements.
– YCor
Jan 4, 2021 at 9:39

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.