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.
Best Answer
The most common thing like this is the near-ring.
I think this question is totally a duplicate of Why is ring addition commutative? but I am reluctant to hammer it.