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

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.

"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."

I 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)