Talk:Finite ring

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This article is suffering from a lack of consistency in the definition of "ring", specifically whether the existence of a multiplicative identity is required. On the one hand it gives examples of rings of order 4 which are not commutative, but on the other hand it claims that every ring of cubefree order is commutative.

Other than the Monthly problem, are there other aspects of the theory of finite rings for which it is fruitful to allow rings without identity? If not, we should follow the wikipedic convention that a ring necessarily contains an identity and state the results accordingly. Plclark (talk) 18:46, 8 August 2009 (UTC)

I am certainly not an expert on finite rings but I think that the complexity of the theory increases when one considers non-unital finite rings (as opposed to unital finite rings). Although it is true that any "ring without 1" can be embedded as an ideal in a unital ring, there are still some aspects of the theory of non-unital rings that are complex in nature. For instance, non-unital rings need not possess maximal ideals (whereas by Zorn's lemma, all unital rings possess maximal ideals). This leads to possible difficulties in defining the Jacobson radical, for instance. However, in the context of this article, no such difficulties can arise.

I did not write the section you quote (so I cannot assess its correctness) but I do agree that if that which you point out is correct, there is some inconsistency. This should be corrected prior to any further progress in this article's quality. Nevertheless, I still feel that this article should study finite rings in their full generality, and therefore consider also non-unital rings. But as I said, I may well be wrong that there is some difference in the theory of non-unital rings to that of unital rings. --PST 02:09, 9 August 2009 (UTC)

Upon a closer look, the necessary phrase "with 1" has always been there in the description of the commutativity theorems based on the order. So there is no mathematical inconsistency. Still, I'm not comfortable with the way the article mixes results about rings with identity together with rings without identity. To me, these are very different objects (of course their theories are different! you pointed out an excellent example about maximal ideals). My question is that are there any contexts in which finite rings with identity occur naturally? Note that the two reference books cited deal only with the case of an identity. Plclark (talk) 03:25, 9 August 2009 (UTC)

Sorry - I meant to say in my last sentence, "...I may well be wrong that there is some difference in the theory of finite non-unital rings to that of unital rings" (I forgot to mention "finite").

With regards to your other point, there seems to be a relatively straightforward theorem that if a finite ring has at least one non-(zero divisor), then it must have a 1. Therefore, the only finite rings without a 1 are those finite rings whose every non-zero element is a zero divisor. --PST 04:40, 9 August 2009 (UTC)

First there are many ring theorists who do not require a ring to have an identity. Secondly there references to various mathematicians should cite the book or article. (remag12@yahoo.com) —Preceding unsigned comment added by Remag12@yahoo.com (talk • contribs) 03:32, 21 July 2010 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Finite ring/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Various aspects of this article can be improved. For instance, a brief summary of finite fields is appropriate, as well as the importance of finite rings in number theory. Also, the article needs further references, and a brief history section. --PST 07:14, 7 August 2009 (UTC)

Last edited at 07:14, 7 August 2009 (UTC). Substituted at 02:06, 5 May 2016 (UTC)

Dangling reference to "Jacobson 1985"

The section Finite ring § Finite field references a publication "Jacobson 1985". The references section, however, does not list such a publication. – Tea2min (talk) 07:42, 2 April 2020 (UTC)

Presumably, that's supposed to be

Basic Algebra. Freeman, San Francisco 1974, Vol. 1; 1980, Vol. 2; 2nd edition, Vol. 1. 1985. 2nd edition, Vol. 2. 1989..

(Taken from Nathan Jacobson § Books. Can somebody please check? Thanks! – Tea2min (talk) 07:46, 2 April 2020 (UTC)