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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.

exchange
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    Do you have common or useful examples of such a thing? If not, that might explain why there's no terminology. – Randall Jul 10 '18 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? – stressed out Jul 10 '18 at 18:12
  • Nearly everything that one can define has some interesting structure. I mean take a group, there are extremely many that are interesting and non-abelian. So it is quite natural to ask whether there are groups that are also monoids but not abelian. – exchange Jul 10 '18 at 18:14
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    See https://math.stackexchange.com/questions/609364/why-is-ring-addition-commutative In particular there’s a link there to https://en.wikipedia.org/wiki/Near-ring It appears the distributive property needs to be adjusted to make this work. – spaceisdarkgreen Jul 10 '18 at 18:15
  • Take a non-abelian group and define an additional operation $\odot$ by $a \odot b=e$ for all $a, b$ in the group (where $e$ is the identity). Done. So what? – Randall Jul 10 '18 at 18:16
  • Thanks for the link. So indeed it is not a stupid question @Randall – exchange Jul 10 '18 at 18:16
  • @Randall, your example might not be interesting but obviously there are many more possibilities. – exchange Jul 10 '18 at 18:17
  • @exchange Yes, it is certainly possible, but I would actually like to see some interesting examples (and I don't think your question is dumb). – Randall Jul 10 '18 at 18:19
  • @Randall, ok I understand. I added a comment to explain why I asked. – exchange Jul 10 '18 at 18:24
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    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.) – Randall Jul 10 '18 at 18:26
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    https://en.wikipedia.org/wiki/Monoid_(category_theory) – Randall Jul 10 '18 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.

rschwieb
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