Started editing the article. I'm afraid it doesn't easily lead itself to being turned into prose.
- I removed the existence of an identity element from the axioms.
- I cut down on the number of examples, rather focusing on the basic examples. If you are looking for examples of rings, I guess knowing that the set polynomials are rings is good enough, and knowing that the set of polynomials with coefficients in a ring i a ring only creates confusion. Same fact with matrices and functions.
- Should we say more about ideals?
- Removed "Basic theorems"
- Cut down on "Construction of new rings"
- History of the subject should be included. (Problem with regards to ring theory page)
Apart from that, I don't know what else should be here, except possibly some things from ring theory, if that is merged in. Maintaining a separate page on ideals could also be discussed. Simen Rustad 13:49, 2 November 2006 (CST)
I put in some examples, but took them out again when I noticed that there was already a section with examples further along in the article. However, I think it would probably be a good idea to have examples sprinkled into the text here and there as I did, or at least introduced earlier in the article. It's easier to imagine what's being talked about if one has at least one example in mind. One way to do this is just to move the "examples" section up to come right after "formal definition" and before "types of rings".
"Conversely, if I is an ideal of A, then there is a natural ring homomorphism from A to A/I such that I is the set of all elements mapped to 0 in A/I." I wonder whether this would make more sense if it ended instead with "...is the set of all elements mapped to 0 by that homomorphism".
"Given the multiplication · in R the multiplication ∗ in Rop is defined as b∗a := a·b. " I would reverse the order of the variables, i.e. I would define it as "a∗b := b·a. "" It seems more natural that way, with the simpler form on the left.
I suggest the following version of this bit; perhaps someone can verify that I have the math right:
- "* The direct sum of a collection of rings is the subring of the direct product consisting of all n-tuples (or infinite-tuples) with the property that rj=0 for all but finitely many j. In particular, if J is finite, then the direct sum and the direct product are identical, but in general they have quite different properties."
Here I've changed "infinite-tuples" to "n-tuples (or infinite-tuples)" because the finite case needs to be covered as well as the infinite case. I added the condition "" to the subscript. I changed "isomorphic" to "identical".
I can't follow the following and I suspect it may be too advanced for this article (or else it requires some definitions and clarification):
- Since any ring is both a left and right module over itself, it is possible to construct the tensor product of R over a ring S with another ring T to get another ring provided S is a central subring of R and T.
Here it would be helpful to have definitions of "module" and "tensor product" available. I notice that with "tensor product" there seem to be three arguments indicated by the prepositions "of", "over" and "with". This seems confusing -- unless maybe the definition of tensor product uses the same three prepositions. If this is kept in the article I think it would be good to slow it down by going through an example, as well as explaining it more fully. --Catherine Woodgold 13:11, 28 April 2007 (CDT)
Example of an ideal?
Would the even integers be an example of an ideal? No identity element. Still, they're an easy concept to grasp, so I think it would be good to mention them as an example while defining ideals and related concepts so that people have something to form a mental image of. For another example of an ideal: if I'm not mistaken, an ideal with an identity element in the set of all 3-by-3 matrices with integer coefficients is the set of all such matrices with all elements in the righthand column and bottom row equal to 0. The matrix with 1,1,0 in the diagonals and zeroes elsewhere is an identity element in the ideal although not in the larger ring.
- The current definition of ideal is wrong. Ideals are not subrings. They're sets that are closed under addition and have the multiplicative-closure (-under-multiplication-by-ring-elements) property described. In particular, any ideal that contains the ring's identity element is automatically the entire ring, by the multiplicative-closure property. Sometimes an ideal can be thought of as a ring with its own identity element (e.g., 2x2 matrices sitting inside 3x3 matrices in the upper-left corner, as you described above), but that's not a subring, where the original ring's identity element must be kept (e.g., Z is a subring of Q). The set of even integers is a great (in fact, fundamental) example of an ideal, as is the set of multiples of any nonzero integer. - Greg Martin 15:06, 1 May 2007 (CDT)
It would be good to mention at least one application of rings and ideals -- perhaps something about solving Rubik's cube or something. (Maybe Rubik's cube only uses groups, not rings.)
A comment in the wikitext says " would be nice to have an example here where the ring and its opposite are genuinely nonisomorphic". I think an example of such an ideal is the set of 3-by-3 matrices with zeroes in the righthand column. (Or 2-by-2 could be used for each of these examples.) --Catherine Woodgold 13:35, 28 April 2007 (CDT)
definition of field?
I added this to the text but am not sure I've included all the conditions for the definition of a field: " A ring in which every element except the additive identity element (0) has an inverse is called a field." Does mulitiplication have to be commutative in a field, for example? --Catherine Woodgold 07:49, 1 May 2007 (CDT)
- Yes. If multiplication is not commutative, you have a division ring. A particularly famous example of a division ring is Hamilton's quaternions. Another, less common, term is skew field. Greg Woodhouse 17:25, 1 May 2007 (CDT)
- OK, I inserted "commutative". I hope I haven't forgotten anything else. --Catherine Woodgold 21:44, 1 May 2007 (CDT)
rethink article structure
Right now this article is looking like a collection of facts. Also (perhaps not coindicentally), it's virtually completely inaccessible to someone who hasn't seen these facts before. (The first several sentences are reasonably good, but the inaccessibility ramps up very quickly.) Although I won't point to any one fact from the current article and swear that it couldn't be fit smoothly into a narrative ... right now the article isn't a narrative, nor an introduction to the subject.
I think we should really rethink the whole structure of the article. What familiar mathematical sets are already examples of rings? What good is thinking about properties of abstract rings? (Answer: it lets us prove theorems about lots of sets at once.) Why did mathematicians define rings to begin with? What's the point of ideals? (Like so much algebra, ideals were developed when people were trying to prove Fermat's Last Theorem! They were originally called "ideal divisors" and were meant to be sets that emulated divisors of numbers, in particular in a way that allowed for unique factorization in what we now call rings of integers in number fields.)
Most importantly, let's ask ourselves the question "what's the most likely reason a person would come to CZ's article on rings in the first place?", and then craft the article to address the answer. - Greg Martin 15:15, 1 May 2007 (CDT)
- Good approaches. Some of those things I may not know the answers to. My thought was to turn it into a narrative by sprinkling examples throughout (and perhaps incidentally deleting the separate example section). Whenever I read a fact or definition, I like to see a (simple) example immediately given to illustrate it. This is probably compatible with what you're suggesting but may not in itself achieve all of it.
- I think we should consider more than one type of reader. I can think of two: One who has seen the word "ring" on another Citizendium math page, thinks "what the heck does that mean?" and follows a link to come here. Another is one who is familiar with rings but doesn't remember one of the axioms or common facts about them and wants to quickly look them up. Why are they called "rings"? --Catherine Woodgold 17:15, 1 May 2007 (CDT)