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I see that a ring is a triple $(R,\cdot,+)$. I am confused by the terms abelian group and semigroup. Does this mean for $x \in R$ and $y \in R$, $x \cdot y$ and $x+y$ are defined?

If so, how is this a ring? When I imagine a ring, I think of some kind of cycle data structure. Like a matrix that is looped shape. Does this sort of thing come into play?

Of course, I should point out that I have not studied ring theory or anything. I just noticed the concept and I find it strange that a "ring" does not seem to be something resembling a loop.

user64742
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1 Answers1

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A ring is (roughly) a set where you can add and multiply the elements together and multiplication distributes over addition.

There are two separate prototypical examples of (commutative) rings:

  1. Take a geometric object, and take all the coordinate functions on it. These can be added or multiplied (pointwise), giving a ring. Depending on what exactly you allow as a "coordinate function," the geometric space can sometimes be recovered purely from the abstract structure of the ring.

  2. Take the rational numbers and throw in a finite list of algebraic numbers, together with everything you can make from them by adding and multiplying. The structure of such rings is important in number theory; for instance, one way of approaching Fermat's Last Theorem is to use the factorization $$x^n + y^n = (x + y)(x + \zeta y)(x + \zeta^2 y) \cdots (x + \zeta^{n-1} y)$$ where $\zeta$ is a primitive n-th root of unity, e.g. $\zeta = e^{2 \pi i / n}$. The way factorization behaves in such a ring is important.

Commutative ring theory essentially exists to bridge these two classes of examples -- people realized that facts and techniques from one area "magically" applied to the other.

The prototypical example of a noncommutative ring is a set of operators on a commutative ring.

The name "ring" itself is a historical accident and it's not worth worrying about what it means; it's a poorly chosen name and should be replaced if anyone could be bothered to coming up with something better.

Daniel McLaury
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  • That final conclusion seems rather hasty. In what way is it an accident and why do you think it is poorly chosen? I can think of any reasons that wouldn't also condemn most other terms used in algebra. – rschwieb Jul 09 '16 at 15:23
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    The motivation for the term appears to be things like cyclotomic numbers, where if you take a high enough power of something it cycles. So the term "ring" basically means "this is an object which may (or may not!) have elements of finite multiplicative degree." I don't really feel that's the defining aspect of a ring. – Daniel McLaury Jul 09 '16 at 15:35
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    At any rate you're not going to get a useful intuition for rings (or fields, or...) by understanding where the terms come from. – Daniel McLaury Jul 09 '16 at 15:37
  • Ok, then we are roughly thinking along the same lines: terms like this are merely not self-explanatory. That's true for so many terms though, and the alternative usually seems to be long names. For things as abstract as rings, it is probably not realistic to expect alternatives that do that nicely. – rschwieb Jul 09 '16 at 18:37
  • Well, I think "algebra" is a very nice term, and of course we could just replace "ring" with "$\mathbb{Z}$-algebra." – Daniel McLaury Jul 09 '16 at 18:57
  • From one point of view, but even that's a bit awkward. For example, in most cases you expect an $R$ algebra to contain a copy of $R$ in its center (but then you have rings without identity.) Secondly, $R$-algebras are usually defined as rings with this extra relationship to $R$, so something would need to be done to avoid circularity. Anyhow... what will you do with groups, fields, modules, integral domains etc? Keeping these 'nicknames', from whatever historical origin, seems to be an unavoidable fate. – rschwieb Jul 09 '16 at 19:19
  • Historical names often stick around for a reason after all: if it is used for a long time and you abandon it for something "better" then you cause confusion in the literature. – rschwieb Jul 09 '16 at 19:21
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    @rschwieb why did you mark this question as duplicate. The other question is not even remotely the same. – user64742 Jul 09 '16 at 23:00