Talk:Ring (mathematics): Difference between revisions

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imported>Aleksander Stos
(chklst)
imported>Catherine Woodgold
(examples)
Line 33: Line 33:
Added some basic history, and merged with [[Ring theory]]. The history section should be fleshed out, however, by someone with more knowledge than me.
Added some basic history, and merged with [[Ring theory]]. The history section should be fleshed out, however, by someone with more knowledge than me.
[[User:Simen Rustad|Simen Rustad]] 13:47, 9 November 2006 (CST)
[[User:Simen Rustad|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''<sup>op</sup> 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 <math>\{R_j\}_{j\in J}</math> is the subring of the direct product consisting of all n-tuples (or infinite-tuples) <math>\{r_j\}_{r_j \in R, j\in J}</math> with the property that ''r<sub>j</sub>''=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 "<math>r_j \in R</math>" 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 (mathematics)|module]] over itself, it is possible to construct the [[tensor product of rings|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.  --[[User:Catherine Woodgold|Catherine Woodgold]] 13:11, 28 April 2007 (CDT)

Revision as of 13:11, 28 April 2007


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)

To learn how to fill out this checklist, please see CZ:The Article Checklist.





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 Rop is defined as ba := a·b. " I would reverse the order of the variables, i.e. I would define it as "ab := 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."

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 "" 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)