User talk:Richard L. Peterson

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Kind Regards, Robert Tito |  Talk  10:50, 29 March 2007 (CDT)

Zero being a divisor

Hi Richard, I modified a little the article divisor and tried to explain my changes on the Talk:divisor page, but unfortunately it didn't work (there seems to be a bug preventing me of creating this talk page). Here is the explanation I intented to put on the talk page.

My textbooks do not require a divisor to be non zero : I usually read that in a -- say commutative -- ring, d is a divisor of a if there is a k such that a=kd, whithout any other requirement. I suppose some authors may exclude the case d=0 to avoid to define the quotient 0/0. I made some changes in the articles in this sense. On the other side, it is true that d=0 is excluded when one wants to define a "divisor of zero" in any ring, and then define an integral domain as a ring without divisors of zero; but anyway, here, "divisor of zero" is a slightly different thing, because it is required this time that k is non zero (else any element of any ring would be a divisor of zero).

Feel free to comment on it and to edit this article again if needed. Best regards, --Sébastien Moulin (talk me) 05:28, 31 March 2007 (CDT)

Your modification is nice, thanks!Rich 15:15, 31 March 2007 (CDT)

update etc on Divisor; Larry Sanger's comment on Divisor discussion

Hi Sebastien, it'll be great to work with someone like you who knows math and has a knack with words!-- I thought your modification of the "zero never a divisor" stuff was very good, in fact, felicitous, but Greg Woodhouse has changed it in a way that I'm also ok with, how about you?-- The other thing is Larry Sanger's comment on Talk about a "plainer language if inexact" def being provided first. He thinks that would be desirable. If you're in favor do you have any ideas for a plainer language definition? On (W.)ikipedia they haven't yet (I think) defined 'divides' formally-- they say "divides evenly" on various prime number pages, which may be good for readers who had math teachers who used those words to mean no remainder or fractional part. I've also seen somewhere on W. "7 divides 42 since 42/7 is an integer", which indeed is plainer than the formal definition. Regards,Rich 19:56, 31 March 2007 (CDT)

Yes, I support the exclusion of 0 amongst possible divisors now, and I agree with Greg Woodhouse's modification (and then with your original definition). I think both versions can be found in litterature, but the case of 0 is so special that it is definitely best to exclude it from the very definition. In fact, the proper way to define divisibility in a ring $R$ would be to define the set of divisors of $R$ as $\Delta =R\setminus \{0\}/U$, where $U$ is the set of inversible elements of $R$ (the quotient is for the equivalence relation defined by $a{\mathcal {R}}b$ if there is an inversible element $u$ so that $a=bu$). On $\Delta$, the divisibility is a partial order that allows to define gcd and lcm properly.

Of course, this formalism is on the opposite of Larry's requirement. And Larry is right: the introduction must be as clear as possible for the non specialist. That's why I fully agree with your proposition about introducing the subject with the notion of division with or without a reminder, with an example like 42/7. It may be a way to stress the importance of euclidean division in arithmetics anyway. Then, we could go to the formal definition (for integers), and then remark that it can be extended to any ring, and give simple examples with polynomials (like $X-1|X^{2}-1$). It could be remarked than the notion of divisibility is also useful in rings without euclidean division (like polynomials of several variables), and so on.

In short, I am in favor of a progressive text, starting at a very accessible level, with integers, and slowly getting more general to encompass the full concept of divisibility in various situations.

Best regards, --Sébastien Moulin (talk me) 17:13, 1 April 2007 (CDT)