Talk:Ring theory

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 Field: Algebra

comment by Tarquin[edit]

Euclidean domain => Principal ideal domain => unique factorization domain => integral domain => Commutative ring.

Is the above a hierarchy of inclusion? If so, use the subset symbol. -- Tarquin

It's inclusion, but it doesn't make much sense to use subset symbol, because there aren't any standard "symbols" for "the class of all Euclidean domains", e.g., unless you want to make up several just for this article. Writing "Euclidean domain contained in PID contained in UFD, etc." is misleading, because it makes it sound like a ED is set-theoretically contained in a PID, contained in a UFD, not the same. Revolver 02:08, 11 Jun 2004 (UTC)

A ring is called commutative if its multiplication is commutative. The theory of commutative rings resembles the theory of numbers in several respects, and various definitions for commutative rings are designed to recover properties known from the integers. Commutative rings are also important in algebraic geometry

Except that the ring article contradicts this and requires commutativity, which adds credence to my belief that this shouldn't be required in the definition. Revolver 02:08, 11 Jun 2004 (UTC)
No, multiplication in rings is not generally required to be commutative. I double-checked this in Herstein's Topics in Algebra as a sanity check. Isomorphic 02:19, 11 Jun 2004 (UTC)
I was wrong, the ring article doesn't require it, it requires unity. My fault. (Although I disagree with unity requirement, as well, that's another matter.) Actually, I just disagree with these universal wikipedia definitions (rather than article by article basis).
There are important examples of rings that do not have a identity. Regardless, and identity can always be formally adjoined by using the adjoint of the forgetful functor from the category of rings with unity to the category of rings.

There is inconsistency with the definition of "Ring" in the main article about rings. As far as I know a ring is assumed to have an identity unless stated otherwise and not the other way around. —Preceding unsigned comment added by (talk) 11:28, 29 May 2009 (UTC)

Vote for new external link[edit]

Here is my site with ring theory example problems. Someone please put this link in the external links section if you think it's helpful and relevant. Tbsmith

Patent nonsense[edit]

Sorry for exaggerating in my edit comment -- the patent nonsense was only in Wikipedia for a little over a day before I reverted it. (I misread the date.) I find it embarrassing, though, that someone who trusts Wikipedia asked me for an explanation of it.—GraemeMcRaetalk 04:48, 21 January 2006 (UTC)

Could use an example[edit]

I don't know much about ring theory, other than that it keeps popping up on wikipedia everywhere... Could someone give some examples of rings and the 2 binary operators?

The integers Z, the rational numbers Q, the real numbers R the complex numbers C all under their ordinary addition and multiplication. Square matrix rings over any of these previous examples are also rings with matrix addition and multiplication. Try to chase some of the links on the ring theory page to arrive at more specific pages: they are likely to have other concrete examples. Rschwieb (talk) 15:25, 14 November 2011 (UTC)

Too Many Math Articles In Wikipedia Suck[edit]

The first two sentences are awful. Repeating the idea that ring theory is the study of rings. Very informative - not. Too much premature jargon. — Preceding unsigned comment added by (talk) 23:54, 17 December 2011 (UTC)

sorting out ring theory and ring (mathematics)[edit]

Discussion started (and will proceed) at Talk:Ring_(mathematics)#Sorting_out_ring_theory_and_ring_.28mathematics.29 Rschwieb (talk) 15:47, 7 February 2013 (UTC)

need a definition[edit]

What is the definition (or what are the axioms) of a ring? Are they clearly stated in this article? I couldn't find them in a quick review. If they are there, please excuse me for asking. However, I think the first section after the intro / lede should say something like, "A ring is a set with two binary operators (typically called "addition" and "multiplication"), satisfying the following: ... ." I think the article might be easier to read and understand if this kind of definition appeared early in the article. Someone with a ring theory book handy should be able to provide this with a reference. Unfortunately, I don't have such handy. Thanks, DavidMCEddy (talk) 03:46, 2 March 2016 (UTC)

There's a separate article at ring (mathematics). However, there's been discussion on whether ring (mathematics) and ring theory should be merged, see e.g. Talk:Ring (mathematics)/Archive 4#merging ring theory into this article and Talk:Ring (mathematics)/Archive 3#New picture, new text. – Tobias Bergemann (talk) 10:48, 2 March 2016 (UTC)