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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.

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    $\begingroup$ Do you have common or useful examples of such a thing? If not, that might explain why there's no terminology. $\endgroup$
    – Randall
    Jul 10, 2018 at 18:12
  • $\begingroup$ 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? $\endgroup$ Jul 10, 2018 at 18:12
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    $\begingroup$ 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. $\endgroup$ Jul 10, 2018 at 18:15
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    $\begingroup$ 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.) $\endgroup$
    – Randall
    Jul 10, 2018 at 18:26
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    $\begingroup$ en.wikipedia.org/wiki/Monoid_(category_theory) $\endgroup$
    – Randall
    Jul 10, 2018 at 18:28

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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.

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