What are the binary operations of a ring?

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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!

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1 Answer

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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.

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