Is there a name for a non-abelian group which is also a monoid?

A ring is usually defined to be an abelian group under addition and a monoid under multiplication.

I wondered whether there is a name for some structure that is simply a (not necessarily abelian) group under some binary operation and a monoid under multiplication.

Is there a name for this? If not, why not?

EDIT: As a response to the comments, I want to add my motivation for the question. Indeed, I do not have a specific example in mind. I asked because I observed that the structure of non-abelian Groups or Rings is much richer than the structure of abelian ones and thus thought there might be a rich world of non-commutative rings with non-commutative addition. And I don't know all the history of mathematics, so I thought maybe someone already researched about this in a deeper way and brought up many examples I could not have thought of. Conversely, there could be a reason why those structures have not been studied yet, e.g. because due to some effect it is algebraically difficult to construct such a thing etc.

• Do you have common or useful examples of such a thing? If not, that might explain why there's no terminology. Jul 10, 2018 at 18:12
• Do you have any particular case of interest in mind? All well-known algebraic structures that I can think of are additive under addition. (rings, modules/vector spaces, etc.) Why should we define something that is not interesting in mathematics? Jul 10, 2018 at 18:12
• See math.stackexchange.com/questions/609364/… In particular there’s a link there to en.wikipedia.org/wiki/Near-ring It appears the distributive property needs to be adjusted to make this work. Jul 10, 2018 at 18:15
• To be fair, there is a name for this sort of thing in any setting. They're roughly called "monoid objects in a category." You're asking about monoid objects in the category of groups, I suppose. (This definition might also force distribution, depending on the category, which you may not want.) Jul 10, 2018 at 18:26
• en.wikipedia.org/wiki/Monoid_(category_theory) Jul 10, 2018 at 18:28