|WikiProject Mathematics||(Rated Start-class, Low-priority)|
Zero vs. Trivial
There seems to be some disagreement as to the definitions of these terms in the literature. The majority of sources I've found use trivial to denote a ring with one element and zero to denote a ring where all products are zero. However some exceptions exist, for example Seth Warner in Modern Algebra and Topological Rings uses the opposite terminology, and MathWorld's zero ring article is just a link to trivial ring which is defined as a ring with one element. In any case the 'a' versus 'the' distinction as used in the current version of this article does not seem to be used in the literature; Warner uses the phrase "a zero ring" and other sources use the phrase "a trivial ring". I couldn't find any sources defining a null ring to be anything other than a ring with one element. I'm going to make changes to this article and possibly the trivial ring article to reflect these results but I wanted to document the issues that came up while trying to determine which terminology is standard.--RDBury (talk) 11:34, 9 March 2010 (UTC)
I propose that the material of this article zero ring be merged into a section of the article pseudo-ring. There is not much material here, and much of it is already present in a section of the pseudo-ring article. Moreover, the usage of the title of this article goes against Wikipedia's convention that rings are unital. Finally, most modern algebra texts by notable authors (Artin, Atiyah and Macdonald, Bourbaki, Hartshorne, Lang's undergraduate algebra, Rotman) use "zero ring" to mean the ring in which 0=1, so it is troublesome to have this material squatting on an article with this title! Ebony Jackson (talk) 23:09, 16 November 2013 (UTC)
- Irrespective of the debate about use of the term "zero ring", I support merging since the object in which the product is always zero is probably largely of interest in the broader context of pseudo-rings, and a section there can adequately cover this concept.
- As to the name debate, I offer no comment due to lack of access to adequate notable sources. —Quondum 23:38, 16 November 2013 (UTC)
Looking at the history of this article, one sees that it originally was a redirect to trivial ring, and only later did someone add the present material (perhaps taking it from PlanetMath, which has an article on this from 2003 predating this one). By the way, Bourbaki mentions both concepts: for Bourbaki, the ring with 0=1 is the "zero ring", and a rng in which every product is zero is a "pseudo-ring of square zero". Ebony Jackson (talk) 00:13, 17 November 2013 (UTC)
Merge "trivial ring" into "zero ring"
I propose to merge the page trivial ring into this page zero ring, and to make the former redirect into this one. The terminology "zero ring" is more common, as evidenced by the most notable references in the subject (as cited in the article zero ring). The only advantage I see to keeping the page at "trivial ring" is that it disambiguates it from the notion of pseudo-ring of square zero, but Wikipedia needs to reflect what is actually used in the current mathematical literature by the most notable mathematicians, so it should be called zero ring. I have placed at zero ring a draft of the material that I think the combined article should contain, including references to justify the claim that this terminology is standard. Ebony Jackson (talk) 16:46, 30 November 2013 (UTC)
- I see no reason to retain Trivial ring – the disambiguation is done by the redirect and the text of Zero ring. —Quondum 02:17, 1 December 2013 (UTC)
Constructions: direct product
Since the direct product is mentioned in Zero ring#Constructions]], should we not also add the following? (I will leave this to those who are familiar with the field.)
- The direct product of the zero ring and any ring is isomorphic to the latter ring.
—Quondum 18:11, 12 January 2015 (UTC)
Contradiction of definition of a prime ideal (w.r.t. the zero ideal) elsewhere?
Prime ideal says (abundantly) that the zero ideal is a prime ideal. This seems to contradict the statement ... the fact that its zero ideal is not prime. (A private opinion: a zero divisor should be defined to include the element of the zero ring; there is too much "playing with words" and too little "using words to describe mathematically regular concepts" in mathematics). —Quondum 05:26, 11 April 2015 (UTC)
@Quondum : I just now came across this via a different route. I this case "its zero ideal" coincides with the entire ring, which is the real culprit. The ring as an ideal in itself should not be considered prime. Emphasizing that it is a zero ideal is a red herring. I went ahead and removed the statement since it did not really add anything. Rschwieb (talk) 18:20, 15 June 2015 (UTC)
- Thanks. English and details of triviality don't mix well. Maybe that's why triviality is often banished. Your removal of the statement is probably the only sensible action here. —Quondum 21:11, 15 June 2015 (UTC)