1

With respect to the usage of the term "universal" in category theory I struggle to see the connection? In terms of elements, if we were to let divisibility represent the morphism, then uniqueness is lost since universal side divisors remain universal side divisors when multiplied by unit. In terms of ideals, then the diagram with inclusion maps represent morphisms doesn't work as seen with the example $(2),(3)\subset\mathbb{Z}$. I have also heard that this term originates from Dummit and Foote's Abstract Algebra course text. Is this true?

And to follow up this original question: Why is it called side divisor? Is there a reason for it? I predict it is because it is different from a regular "divisor" since the elements it'll divide will need to be "transposed to another side" (the choice of added unit/zero element can be thought of as a "side").

Please help. Thank you all very much!

RobPratt
  • 50,938
DoubleA Batteries
  • 13
  • 4
  • 2
    "Universal" here is not in the sense of a universal property of category theory, but in the sense of "encompassing everything". It divides every element... up to a unit. – Arturo Magidin Jun 05 '24 at 20:22
  • When you study analysis, don't try to find a connection between category theory and the Baire category theorem: there is none. That work of Baire was decades before category theory was developed. – KCd Jul 28 '24 at 08:43

1 Answers1

0

A nonzero nonunit $\,d\,$ is called a side divisor of $\,a\,$ if $\,d\mid a-u\,$ for $\,u\,$ some unit (or $0),\,$ and a universal side divisor if it is a side divisor of all elements in the ring, i.e. every element is congruent to a unit (or $\,0)\,$ modulo $\,d$.

As for its raison d'etre, the universal side divisors provide a convenient way to show some domains are not Euclidean, e.g. see my comments here. They correspond to the second stage of minimal "remainder values" in the general Motzkin transfinite Euclidean algorithm construction - which yields a way of testing if a domain is Euclidean (e.g. see Lemmermeyer's paper A Survey on Euclidean Number Fields).

Bill Dubuque
  • 282,220
  • Hello sir, thank you for your help. This is really good. Im sorry to ask follow ups but do you know where the term "side divisor" comes from, and "universal side divisor"? I saw a post on here say it was actually from Dummit and Foote (as I suspected) but your answer seems to be different than his, and his answer is gone?? – DoubleA Batteries Jun 05 '24 at 21:29
  • It is mentioned in D&F but the name may be older. It might be due to Hendrik Lenstra. – Bill Dubuque Jun 05 '24 at 21:32
  • @Arturo I have a vague recollection of Lenstra using it but I cannot recall where (it could be as old as circa 1982 when Lenstra sent improvements on the original APR primality test we were developing at MIT). I also recall Lenstra using "side exit" in algorithms (Henri Cohen mentions such in his book on computational number theory, section 2.3.4). $\ \ $ – Bill Dubuque Jun 05 '24 at 22:09
  • The terms "side divisor" and "universal side divisor" are not at all due to D&F or Lenstra. They go all the way back to Motzkin's paper on the Euclidean algorithm in 1949: https://www.ams.org/journals/bull/1949-55-12/S0002-9904-1949-09344-8/S0002-9904-1949-09344-8.pdf. – KCd Jul 28 '24 at 08:19
  • @DoubleABatteries where on MSE did you see anyone claim the weird "side divisor" terminology originated with D&F? – KCd Jul 28 '24 at 08:22
  • @KCd Thanks for jogging my memory on the history of the term, – Bill Dubuque Jul 28 '24 at 08:23
  • I first came across the "side divisor" lingo years ago in Jack Wilson's paper on a principal ideal ring that's not Euclidean in the Raymond W. Brink Papers on Algebra, and Wilson cites Motzkin's paper. That term made no sense, so I checked Motzkin's paper (it was the '90s, so this meant a trip to the physical library) and he doesn't explain what is "side" about side divisors. So annoying. – KCd Jul 28 '24 at 08:38
  • @KCd I suspect many are puzzled by the terminology. I think I first saw it in some papers that Lenstra sent me when I was an MIT undergrad working with Adleman and Rumely on the APR primality test. My fuzzy memory seems to have conflated it with Lenstra's use of "side exit", though I must have later seen it while perusing Motzkin's paper. $\ \ $ – Bill Dubuque Jul 28 '24 at 08:50