Thread: Which semigroups are multiplicative semigroups of a ring?

I was wondering if there were any results on this. To which semigoups (not necessarily finite) can I add an additive operation so that the resulting structure would be a ring? (I think this is the right forum for this question. Please correct me if it's not.)

Originally Posted by ymar

I was wondering if there were any results on this. To which semigoups (not necessarily finite) can I add an additive operation so that the resulting structure would be a ring? (I think this is the right forum for this question. Please correct me if it's not.)

Hmm...not sure! However, not all semigroups can be made into rings. For example, there are only 2 non-commutative rings of order 4 (think about what group must be the underlying one, and go from there). However, there are a lot more non-commutative semigroups of order 4! (68, I believe. See here.)

Yes, I knew there were more semigroups than rings. For the number of all non-isomorphic semigroups with n elements, see A027851 - OEIS. For the number of finite rings, see http://oeis.org/A027623.

Originally Posted by ymar

I was wondering if there were any results on this. To which semigoups (not necessarily finite) can I add an additive operation so that the resulting structure would be a ring? (I think this is the right forum for this question. Please correct me if it's not.)

It seems this was studied first in the last 60s-early 70s,

On Semigroups Admitting Ring Structure

On Semigroups Admitting Ring Structure II

but has been studied as recently as 2010,

Some Results on Semigroups Admitting Ring Structure

Interestingly, one cannot axiomatise them...

Ha! Great, thanks! :-)

Originally Posted by ymar

I was wondering if there were any results on this. To which semigoups (not necessarily finite) can I add an additive operation so that the resulting structure would be a ring? (I think this is the right forum for this question. Please correct me if it's not.)

Good question!