Talk:Divisibility (ring theory)

Suggestions/comments

I think "Divisibility (ring theory)" is a better title for this article. Otherwise, it sounds like it's about Divisor (algebraic geometry); and this article is probably going to be more about divisibility in rings than specifically about a divisor of an element. Also, this is somewhat of a content fork of Integral domain#Divisibility, prime and irreducible elements, so that should be tidied up. Finally, how often is divisibility studied in non-integral domains? RobHar (talk) 03:44, 9 September 2011 (UTC)

That's good point. ("Divisor" in algebraic geometry didn't occur to me when I imported the article from citizendium.) I think what we really need is Divisor theory. But maybe that's only for domains. I do think divisibility is studied in general; in particular, polynomials and investigating the extent to which Euclid's lemma holds. In any case, I'm moving the article. -- Taku (talk) 15:45, 12 September 2011 (UTC)

Fixing internal inconsistency in article: which definitions are notable?

This article says

A nonzero element b of a commutative ring R is said to divide an element a in R (notation: $b\mid a$) if there exists an element x in R with $a=bx$. We also say that b is a divisor of a, or that a is a multiple of b.

whereas Integral domain#Divisibility, prime_elements, and irreducible elements says

If a and b are elements of the integral domain R, we say that a divides b or a is a divisor of b or b is a multiple of a if and only if there exists an element x in R such that ax = b.

It seems clear to me that there are at least two definitions in general notable use, and I'll assume these:

(a) b | a if a = bk for some k

(b) if b ≠ 0, b | a if a = bk for some k

(c) b | a if a = bk for some k and b ≠ 0

(d) b | a if a = bk for some k ≠ 0 and b ≠ 0

... other variations are possible.

The difference is essentially as follows:

With (a), 0 | 0 is true. b | 0 is true for any b.

With (b), 0 | 0 is undefined. b | 0 is true for any b ≠ 0.

With (c), 0 | 0 is false. b | 0 is true for any b ≠ 0.

With (d), 0 | 0 is false. b | 0 is false for any b that is not a (nonzero) zero divisor.

This article is self-contradictory, in that it gives a definition equivalent to (b), but the Properties section then makes statements that are only true with definition (a). In general it seems to me that statements of theorems become more complicated if other than definition (a) is used, but it is not for me to dictate to notable sources. There seems to me to be no utility in excluding zero elements as zero divisors, with the following exception: a zero divisor is only interesting if it is defined as a divisor of zero with definition (d). It seems to me that we must give weight to more than one definition of divisor, taking care to distinguish (b) and (c). Since (d) seems only to have utility to define a particular useful concept, and this does not seem to be helpful otherwise and I'm guessing is not notable, I'd suggest leaving out this case. — Quondum 21:06, 10 September 2013 (UTC)

 Done - just noting that this has been remedied. —Quondum 20:19, 4 March 2014 (UTC)

Notation for principal ideal

This article uses the notation (a) and links to principal ideal. One has to infer that they are referring to the same thing, since the linked article does not use or define the notation. A nutshell definition here would be appropriate. Also, one may have a left, right or two-sided principal ideal (all the same for a commutative ring), which is a little confusing given that this article does not firmly commit to being within the framework of commutative rings. It would be nice if it actually defined divisors in the non-commutative context if this use is notable, even if the dominant use might be in the commutative case. —Quondum 18:20, 16 November 2013 (UTC)

Shouldn't left divisor and right divisor be defined

The two redirects Left divisor and Right divisor point to this article. The concepts make sense in a general ring (and indeed in any magma). Should they not be defined here? —Quondum 21:53, 4 March 2014 (UTC)

 Done —Quondum 18:17, 20 March 2014 (UTC)

Requested move 2 April 2017

The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review. No further edits should be made to this section.

The result of the move request was: not moved. Additional changes not requiring the technical abilities of an administrator of a page mover can be done per WP:BOLD or further discussion outside of this RM. (non-admin closure) TonyBallioni (talk) 22:37, 9 April 2017 (UTC)

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Divisibility (ring theory) → Divisibility – Currently "divisibility" redirects to "divisibility rules". But this article should be the primary topic of divisibility. GeT RiGhT (talk) 02:04, 2 April 2017 (UTC)

• I'm not sure divisibility rules (via redirect) is the proper content to see at divisibility, but I oppose having Divisibility (ring theory) there. The ring-theory article is very technical and deep math theory for something that is an everyday lay-public and school-student idea. It may be true that the (ring theory) article is the fundamental article we currently have on the topic of divisibility, with the ...rules article being an application, but I think having (ring theory) be the PRIMARYTOPIC does a disservice to the vast majority of our readers. Instead, I propose divisibility be a Wikipedia:Set index article as PRIMARYTOPIC, which introduces the idea of what it means to be divisible by something, with then links to the mathmatically technical (ring theory) and the ...rules applications articles. DMacks (talk) 20:15, 9 April 2017 (UTC)

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The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a move review. No further edits should be made to this section.

Defining divides

Statements along the lines of a divides b are common, but are not included in the definition. This does show up under divisor, but it seems harmless enough to include here as well. 47.142.151.211 (talk) 20:49, 2 July 2017 (UTC)

In commutative ring this is not a problem, but otherwise one needs to say "a left-divides b", etc. With all the variations already present, giving possible variants of how to express each becomes cumbersome. Maybe we should leave inference of verb forms to the reader? —Quondum 19:01, 26 September 2023 (UTC)