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I'm just reading through the definition of a ring and it says that "A ring is the triple (R,+,•) where R is a set and +,• are binary operations".

Does • imply multiplication since for R to be a ring we require associativity of multiplication and the existence of a multiplicative inverse?

Or can it be another binary operation yet those two conditions still hold (despite multiplication not being a binary operation)?

Thank you!

BigWig
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    +,• are just suggestive symbols for these binary operations, you might as well use a square and a blue triangle. The important thing is that these binary operations have the properties postulatd thereafter – Hagen von Eitzen May 06 '18 at 07:36
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    Incidentally, we don't require the existence of multiplicative inverses in a ring. – Eric Wofsey May 06 '18 at 07:36

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The operations $+$ and $\cdot$ can be any binary operations on the set $R$, as long as they satisfy all the ring axioms. They don't have to be the operations we normally call "addition" and "multiplication" on $R$, if such operations exist. However, in the context of the ring $(R,+,\cdot)$, we usually refer to $+$ as "addition" and $\cdot$ as "multiplication". So when we speak of "associativity of multiplication", for instance, we are actually talking about associativity of the operation $\cdot$, whatever it happens to be.

Eric Wofsey
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  • Ahhh! okay thank you! I was under the impression the addition was a must (in terms of how we think about it in the reals) - that was just my mistake for misinterpreting my lecture notes. Thank you very much! – BigWig May 06 '18 at 07:40
  • @ericWofsey when the binary operations satisfy all the ring axioms, is the order of binary operations in ring indication important ? For example, if $(R,&,#)$, then we say that $R$ forms an abelian group with respect to $&$, but if $(R,#,&)$, then we say that $R$ forms an abelian group with respect to $#$. So, the first written binary opeartion symbol is responsible for being abelian group with respect to itself, and the second written operation is responsible for being closed under associavity and satisfy the disributive law – Not a Salmon Fish Oct 09 '23 at 16:54
  • @NotaSalmonFish: Yes, the order is important: a ring is defined as an ordered triple satisfying certain axioms, so the order of the entries in the triple matters. – Eric Wofsey Oct 09 '23 at 19:11
  • @EricWofsey thanks for replying – Not a Salmon Fish Oct 09 '23 at 20:03
  • @EricWofsey can you please look at this question. I think the OP is totaly right, but some others says that the order of binary opearations in not very important. I will delete this comment after you see it – Not a Salmon Fish Dec 21 '23 at 19:30
  • @NotaSalmonFish: It's a matter of convention. You can define a ring to be a triple in whichever order you prefer, as long as you then use that ordering consistently. In practice, people might not be completely consistent about it but everyone understands what they intend so it's not a serious problem. – Eric Wofsey Dec 21 '23 at 20:15