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:



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


