As I’ve studied math and physics as a hobby I’ve always been curious as to why things were called what they were called, e.g. elliptic integrals, self adjoint, unitary transformations etc. Finding the answer usually only entailed some Googling, but despite quite a bit of searching I’ve never been able to find out why a ring is called a ring. So I gave up and just chalked it up to stupidity.
However, yesterday while reading Roger Penrose’s The Road to Reality I stumbled on the following:
So! It seems this isn’t my fault at all—it’s the Mathematicians fault, and Roger and I both think it’s probable that these guys will rot in Hell for this outrage.
Does anyone know what neither Roger nor I know? Why this entity is called a Ring?
The word “Ring” is also used in German. According to one of my mathematics professors and one of our textbooks it is derived from an relatively obscure meaning of the word “Ring” in German: A union of closely connected things.
Even if this meaning is rare, it can be found in some contexts.
The most common use today except in mathematics is in the names of some organizations.
Probably the English “ring” has a similar meaning - Or the German term was used like a few others (e.g. Eigen-whatever.)
Of course that answer’s off the internet, so it’s probably a lie spread by Hilbert’s estate in order to further their lawsuit against the Noether estate, and gain control of the nickel royalty that mathematicians pay each time they use the word ‘ring’.
According to Mathworld, Hilbert first used the term “Zahlring” (or number ring), to describe a particular type of ring, written as Z[3rd root of 2], that is the smallest ring containing the integers (Z), and the 3rd root of 2. The symbolic representation is given on the website. The term ring is, according to them, a short form of Zahlring, and in germany they now use the term Ring as well.
They comment that “By successively multiplying the new element [the 3rd root of 2], it eventually loops around to become something already generated, something like a ring”. This seems to suggest this looping is where the name came from, but it isn’t that clear.
I would give a cite from Wikipedia, but Squink already has, as www.absoluteastronomy.com as the article cited, and seemingly all the encylopaedia content is taken (under the GFDL license), from there.
Ok thanks folks. Now how about this one: When it has an application in physics I seem to be able to grasp abstract math reasonably well. But when it’s just math for math’s sake I just don’t see anything but strange symbols.
So why are rings so important? Why do they have their own name? The axioms they obey seem pretty pedestrian so what’s going on. How about something that’ll give all of us non mathematicians a clear grasp of what pure math is all about. That shouldn’t be so hard should it?
Rings are important because a lot of structures satisfy the ring axioms. You could prove the same things over and over again for each structure, or you could prove it once for a general ring and be done with it. Mathematicians are nothing if not just the least bit lazy.
Thanks ultrafilter that makes sense. How about giving me an example of something that doesn’t meet the axioms for a ring. Preferably something fairly common.
That’s a pretty wide request. A square doesn’t satisfy the axioms of a ring. Perhaps that’s silly, but you have to narrow your question a bit. Let’s assume that you meant to ask for a set with some operation(s) that doesn’t satisfy the ring axioms. The set of symmetries of the square doesn’t satisfy the ring axioms: There is only one binary operation, composition, rather than the two that a ring requires.
So, let’s narrow further to sets with two binary operations. Many of the examples one sees do satisfy the ring axioms, since they describe a simple collection of requirements for the object to be well-behaved. (Pace you algebraists studying Lie algebras and the like.) But that doesn’t mean there aren’t common examples that aren’t rings. One that comes to mind is the set of nonnegative integers with addition and multiplication. All the ring axioms are satisfied except that there are, in general, no additive inverses (no negatives). There are plenty of other examples.
This, I think, shows one of the main problems between mathematicians and non-mathematicians. If I would have known enough to ask the question the correct way then I probably wouldn’t have had to ask the question in the first place.
That isn’t one of the axioms that Penrose lists for rings. I’m certainly not saying you’re wrong but what are these two binary operations, and where does it say there must be two?
The set of symmetries of the square (i^0, i^1, i^2, i^2 and i^4 back) comprise a group I believe. Is there a relation between a ring and a group?
To me this seems silly. Of course there’s no additive inverse-- you specifically excluded them.
A better example: the octonions (pairs of quaternions in the same way that quaternions are pairs of complex numbers) are not a ring because their multiplication is not associative.
A ring is an algebraic structure such that it is a group under one of its binary operations and a monoid under the second one. (The elements of a monoid don’t necessarily have an inverse; those who do are called “units”.) There are also the properties of distributivity of the second binary operation over the first one: a(b+c) = ab + ac and (a + b)c = ac + bc.
Yes, the first binary operation is generally called the “addition” and the second one the “multiplication”, but that’s just a convention and they don’t need to be defined as the operations over the reals that we know about. They can been anything that respects the axioms.
Topologist’s example of a structure that follows most of the ring axioms but not all is a good example, I think. Of course, it’s not a ring because he excluded the negative numbers, but you can look at this structure and see what happens when not all axioms are respected.
Grrrrrrr … I’ve been trying to google on cosmic rings but I obviously have the wrong term here.
Can someone provide the correct term for a ring that is one molecule in length and diameter yet of indeterminate size?
Supposedly, and I got this from reading some stuff written by Professor Gregory Benford, these objects, whatever they are, are left over from the hypotheticl big-bang that created the universe.
Now. The question is:
Would this type of object ALSO be considered a ring in the clasical sense of the term?
Rings are structures where (basically) you can add, subtract, and multiply, but not generally divide. There are lots of such structures out there. Integers, for instance, or real-valued functions (arbitrary, continuous, differentiable, smooth…) on a topological space. Ring theory lets you talk about all these at once in general, the same way high school algebra lets you talk about whole classes of number equations (with variables).
Really, though, I’m of the opinion that you don’t really get mathematics unless you can appreciate ars gratia artis.
I’d just like to thank you, Mathochist, for your lucid and enthusiastic explanations of mathematical topics on here. As a math major, I’ve desperately tried to stay current, and reading your lengthy posts on the SDMB here has been a huge help.
