Talk:Quasiregular element

Definition and article title

I have a few comments to make. First, isn't the term "quasi-regular" (as opposed to "quasi-regular") more popular and standard? Second, according to the definition for a non-unital ring, x is quasi-regular if and only if $1+x$ is quasi-regular (not $1-x$). So, there seems to be a little consistency issue (which is probably not serious.) -- Taku (talk) 12:12, 6 October 2009 (UTC)

With regards to a preference of "quasi-regular" over another term, the first sentence of your post appears to state it twice. With regards to the second aspect, you are right. Most authors of noncommutative ring theory refer to "quasi-regularity of x" as the assertion that $1+x$ is invertible. I have specifically chosen that it should refer to $1-x$ being invertible. Personally, I find this somewhat more natural than the other alternative. For instance, in analogy with the Jacobson radical, suppose x is in some maximal (proper) ideal of a ring; then $1-x$ cannot be in this proper ideal. Likewise, $1+x$ cannot be in this ideal, but for this you must consider the additive inverse of x, and apply the same logic as before. However, this requires "one more logical step". The essential ideas I have considered undermine the characterization of the Jacobson radical (of a unital ring) in terms of quasi-regularity. --PST 00:49, 7 October 2009 (UTC)

Sorry, I meant to suggest "quasi-regular" (with hyphen) might be a better name than "quasiregular" (without hyphen. As for the second, unfortunately, it's not that simple. Remember how eigenvalues work. "quasi-regular" "intuitively" means that -1 is not an eigenvalue (not 1), and so 1+x being quasi-regular isn't the same as 1-x being quasi-regular. This is why we get x^2 is quasi-regular if and only if -x is quasi-regular. Confusing, yes. But we're really not allowed to change definition.... -- Taku (talk) 12:53, 8 October 2009 (UTC)

If you have access to a library in the imminent future, perhaps you would like to have a look at a certain book by Martin Isaacs (cited in the reference section of the article's page). In this book, Isaacs defines quasiregularity in terms of whether $1-x$ has an inverse. Although I personally would not consider this book as a primary source on noncommutative ring theory (despite it being an excellent book should one wish to learn finite group theory or field theory), I think that in subjects other than finite group theory, Isaacs tends to follow the style of other authors of noncommutative ring theory such as Herstein (although, in this case, Herstein defines quasiregularity in terms of whether $1+x$ has an inverse). Therefore, I suspect that there is a good book on noncommutative ring theory which follows the same conventions as Isaacs (although I cannot prove it). Maybe I will have a look in the library soon...

With regards to choosing "quasi-regular" over "quasiregular", I think that this is similar to choosing "non-commutative ring" over "noncommutative ring" (which was recently considered at WP Mathematics). If you like, we can raise this issue (and the above issue) there, but I personally feel that we should work on the article, and worry about these issues later (by the way, I appreciate the work you are doing on this article - I did not think anyone cared about quasiregularity...). --PST 01:13, 14 October 2009 (UTC)

Seems to me there's a bigger problem. If we take R=Z, the ring of integers, then by the definition of the first paragraph, 2 is quasi-regular, since 1-2 is invertible. But by the supposedly equivalent definition for non-unitary rings, it's not, since there is no y such that 2y + 2 + y = 0 (-2/3 not being in Z). I'm guessing you may want xy - x - y rather than xy + x + y. I think it would also improve readability if you either used x in the first paragraph or r in the second. —Preceding unsigned comment added by 69.17.114.80 (talk) 18:51, 15 October 2010 (UTC)

Properties

One of the properties states that an idempotent element cannot be in the Jacobson radical "because idempotent elements cannot be quasiregular." Is this really true? In the rational numbers, considered as a ring, 1 is an idempotent with quasi-inverse -1/2. Perhaps this is related to your choice of using "1 - r is invertible" as opposed to "1 + r is invertible." 173.31.222.232 (talk) 13:11, 23 July 2010 (UTC)

Sign convention

This article has many contradictions based on an inconsistent sign convention. Isaacs and Jacobson use the $1-x$ and $x\circ y=x+y-xy$ convention, while Polcino & Sehgal use the $1+x$ and $x\circ y=x+y+xy$ convention. While in some sense the sign convention makes no substantive difference, it certainly makes a big difference when approximately half the article uses one convention and the other half uses the other. It seems like both conventions must be mentioned, and some simple way to distinguish them should also be used so that the results can be stated clearly. JackSchmidt (talk) 19:58, 25 July 2013 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Quasiregular element/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.

I do not think that much more useful information can be added to this article which does not repeat properties already given at Radical of an ideal or Jacobson radical. Thus, perhaps more references can be added, as well as examples where the notion of quasiregularity proves to be computationally convenient. Concrete examples of quasiregularity may also be worthy to include. --PST 13:25, 5 July 2009 (UTC)

Last edited at 13:25, 5 July 2009 (UTC). Substituted at 02:32, 5 May 2016 (UTC)