Talk:Ring (mathematics)

Article Checklist for "Ring (mathematics)"

Workgroup category or categories	Mathematics Workgroup [Categories OK]
Article status	Developing article: beyond a stub, but incomplete
Underlinked article?	Yes
Basic cleanup done?	Yes
Checklist last edited by	--AlekStos 12:57, 17 April 2007 (CDT)

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Notes

Started editing the article. I'm afraid it doesn't easily lead itself to being turned into prose.

Some points:

• 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)

Added some basic history, and merged with Ring theory. The history section should be fleshed out, however, by someone with more knowledge than me. Simen Rustad 13:47, 9 November 2006 (CST)

examples, etc.

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 R^op 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 $\{R_{j}\}_{j\in J}$ is the subring of the direct product consisting of all n-tuples (or infinite-tuples) $\{r_{j}\}_{r_{j}\in R,j\in J}$ with the property that r_j=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 "$r_{j}\in R$" 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.

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)