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A ring is a set $S$ together with two operations: $*$ and $+$. The operations in question have the following properties: $*$ distributes over $+$, $(R,+)$ forms an Abelian group and $S$ is closed under both operations. That's for sure. There's also associativity of $*$ - some sources I've used list it as another axiom for rings, and I have thought this was the case, but I've just encountered a notion of "non-associative ring" - apparently, such objects exist and there are theorems about them.

Why are rings often defined to be associative, if mathematicians work on non-associative objects which are too called "rings"?

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    $\begingroup$ It would be beneficial if you could provide a reference to a paper/book/... that allows 'rings' to be non-associative. (I'm not questioning whether such a reference exists, but I believe that there might be some insight to be gained.) $\endgroup$ – Stefan Mesken Dec 6 '16 at 18:55
  • $\begingroup$ Because the real numbers and composition are associative. This is where the idea of rings came from. $\endgroup$ – Jacob Wakem Dec 6 '16 at 19:08
  • $\begingroup$ Associativity is an important property of function composition, which you can often use to represent associative objects. Associativity is also built into the axioms of arrows in category theory. That is why associativity is so pervasive. $\endgroup$ – rschwieb Dec 6 '16 at 22:05
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I think the correct conclusion you can draw is that a "non-associative ring" is not a ring.

From this you might wish to draw a linguistic corollary, that "non-associative ring" is bad terminology. I have to say, though, that I kind of like it, even though it is an abuse of language, because it conveys the idea perfectly.

Some mathematical terminology exhibits more sensitivity about this issue. For instance, in group theory, a "monoid" is a mathematical object which might be called a "group without inverses" if someone wished to be linguistically sloppy, or if someone just wanted to convey the idea perfectly.

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Why are rings defined to have operations $+$ and $\times$, when often mathematicians work with "rings" without those operations?

Simply because such operation-less rings are extremely useful in their own right, so we give them a name all for themselves (namely "set").

However, non-associative rings aren't anywhere near as widely studied as rings are, so we rely on the context supplied by the author to tell us when a ring is intended to be non-associative. (In my experience, that's literally never, but it seems that's just because I'm inexperienced.)

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  • $\begingroup$ Non-associative rings, like Lie rings, are certainly (almost) as widely studied as rings. $\endgroup$ – Dietrich Burde Dec 6 '16 at 19:15
  • $\begingroup$ Then I take it back :) $\endgroup$ – Patrick Stevens Dec 6 '16 at 19:16
  • $\begingroup$ No problem :) :) $\endgroup$ – Dietrich Burde Dec 6 '16 at 19:16

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