Re: GCD(0,0)

On Sat, 31 Dec 2005 04:23:01 -0500, quasi <quasi@xxxxxxxx> wrote:

>On Sat, 31 Dec 2005 04:07:38 -0500, quasi <quasi@xxxxxxxx> wrote:
>
>>On Sat, 31 Dec 2005 03:48:59 -0500, quasi <quasi@xxxxxxxx> wrote:
>>
>>>On Fri, 30 Dec 2005 17:16:54 -0500, quasi <quasi@xxxxxxxx> wrote:
>>>
>>>>On 30 Dec 2005 01:13:33 -0800, "Proginoskes" <CCHeckman@xxxxxxxxx>
>>>>wrote:
>>>>
>>>>>
>>>>>quasi wrote:
>>>>>> On Thu, 29 Dec 2005 21:45:41 -0500, quasi <quasi@xxxxxxxx> wrote:
>>>>>>
>>>>>> >On Thu, 29 Dec 2005 21:29:52 -0500, quasi <quasi@xxxxxxxx> wrote:
>>>>>> >
>>>>>> >>On Thu, 29 Dec 2005 20:58:08 -0500, quasi <quasi@xxxxxxxx> wrote:
>>>>>> >>
>>>>>> >>>On 29 Dec 2005 19:59:26 -0500, hrubin@xxxxxxxxxxxxxxxxxxxx (Herman
>>>>>> >>>Rubin) wrote:
>>>>>> >>>
>>>>>> >>>>In article <1135888209.029280.145500@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
>>>>>> >>>>Leroy Quet <qqquet@xxxxxxxxxxxxxx> wrote:
>>>>>> >>>>>
>>>>>> >>>>>But what is, if there is any defined value,
>>>>>> >>>>>
>>>>>> >>>>>GCD(0,0)?
>>>>>> >>>>>
>>>>>> >>>>>Or is GCD(0,0) simply undefined, like 0/0?
>>>>>> >>>>
>>>>>> >>>>When it comes to common divisors, since everything
>>>>>> >>>>divides 0, and otherwise an integer never divides
>>>>>> >>>>a smaller integer, for this purpose, 0 is the
>>>>>> >>>>greatest common divisor of 0 and 0.
>>>>>> >>>>
>>>>>> >>>> The ordering is that of divisibility.
>>>>>> >>>
>>>>>> >>>So you proclaim.
>>>>>> >>>
>>>>>> >>>Let's see what the books have to say.
>>>>>> >>
>>>>>> >>Ok, I'll start it off with Sierpinski ...
>>>>>> >>
>>>>>> >>Sierpinski, Elementary Theory of Numbers, 1964
>>>>>> >>
>>>>>> >>Sierpinski defines GCD for any nonempty set S of integers (possibly
>>>>>> >>infinite) provided at least one element of S is nonzero.
>>>>>> >>
>>>>>> >>quasi
>>>>>> >
>>>>>> >David R. Wilkins
>>>>>> >Topics in Number Theory
>>>>>> >Course Notes, Trinity College (Dublin), 2005
>>>>>> >
>>>>>> >Definition:
>>>>>> >
>>>>>> >Let a_1, a_2, ..., a_r be integers, not all zero. A common divisor of
>>>>>> >a_1, a_2, ..., a_r is an integer that divides each of a_1, a_2, ...,
>>>>>> >a_r. The greatest common divisor of a_1, a_2, ..., a_r is the greatest
>>>>>> >positive integer that divides each of a_1, a_2, ..., a_r. The greatest
>>>>>> >common divisor of a_1, a_2, ..., a_r is denoted by (a1, a2, ..., ar).
>>>>>> >
>>>>>> >quasi
>>>>>>
>>>>>> PlanetMath.org
>>>>>>
>>>>>> greatest common divisor
>>>>>>
>>>>>> Let a and b be given integers, with at least one of them different
>>>>>> from zero. The greatest common divisor of a and b, denoted by
>>>>>> gcd(a,b), is the positive integer d satisfying
>>>>>>
>>>>>> 1. d | a and d | b
>>>>>>
>>>>>> 2. if c | a and c | b then c | d
>>>>>
>>>>>MathWorld:
>>>>>
>>>>>The greatest common divisor GCD(a,b) of two positive integers a and b,
>>>>>sometimes written (a,b), is the largest divisor common to a and b. [and
>>>>>the link to "divisor" says:] A divisor of a number n is a number d
>>>>>which divides n (written d|n), also called a factor.
>>>>>
>>>>> --- Christopher Heckman
>>>>
>>>>This next one is from a ring theorist ...
>>>>
>>>>Robert Gilmer
>>>>Multiplicative Ideal Theory
>>>>1972
>>>>
>>>>Let R be a [commutative] ring with identity.
>>>>
>>>>If a_1, ..., a_n are nonzero elements of R, an element b of R is
>>>>called a greatest common divisor of a_1, ..., a_n (written b =
>>>>GCD(a_1, ..., a_n) if b divides each a_i, and any common divisor of
>>>>the elements a_i divides b.
>>>>
>>>>Notes: Robert Gilmer is making the definition applicable to any
>>>>commutative ring with identity so he's definitely aiming for
>>>>generality. Od course, Gilmer is perfectly aware of what a PID is. The
>>>>fact that he didn't extend the definition to allow GCD(0,0) should
>>>>tell you that it's not important enough to need to extend it.
>>>>
>>>>quasi
>>>
>>>Joseph Landin
>>>Algebraic Structures
>>>1989
>>>
>>>Definition:
>>>
>>>Suppose a<>0 and b<>0. An integer d>0 is the greatest common divisor
>>>(g.c.d.) of a and b if and only if
>>>
>>>(a) d is a common divisor of a and b, and
>>>
>>>(b) if c | a and c | b then c | d.
>>>
>>>quasi
>>
>>Gareth A Jones, Josephine M Jones, J M Tyrer-Jones
>>Elementary Number Theory
>>1998
>>
>>Definition
>>
>>If d|a and d|b we say that d is a common divisor (or common factor) of
>>a and b; for instance 1 is a common divisor of any pair of integers a
>>and b. If a and b are not both 0, then Exercise 1.3(d) shows that no
>>common divisor is greater than max(|a|,|b|), so that among all their
>>common divisors there is a greatest one. This is the greatest common
>>divisor (or highest common factor) of a and b; it is the unique
>>integer d satisfying
>>
>>(1) d|a and d|b (so that d is a common divisor)
>>
>>(2) if c|a and c|b then c<=d (so that no common divisor exceeds d).
>>
>>However, the case a=b=0 has to be excluded; every integer divides 0
>>and is therefore a common divisor of a and b, so there is no common
>>divisor of and b in this case.
>>
>>When it exists, we denote the greatest common divisor of a and b by
>>gcd(a,b), or simply (a,b). This definition extends in the obvious way
>>to any set of integers (not all 0).
>>
>>quasi
>
>G. H. Hardy & E. M. Wright
>
>An Introduction to the Theory of Numbers, 5th Ed.
>1980
>
>We define the highest common divisor of two integers and b, not both
>zero, as the largest positive integer which divides both a and b; and
>write d = (a,b).
>
>Note: Surely a definition by Hardy & Wright should carry some weight,
>no?
>
>quasi

My sense is that Serge Lang was not the kind of mathematician who
could have been coerced to accept a mainstream definition if he felt
an alternate definition was superior. In my view, he was a more of a
leader than a follower in the world of math.

So let's see what Lang had to say ...

Serge Lang
Algebra, 3rd Ed.
2002

Let A be an entire ring and a,b in A, ab <> 0.

We say that a divides b and write a|b if there exists c in A such that
ac=b.

We say that d in A, d<>0 is a greatest common divisor (g.c.d) of a and
b if d|a, d|b, and if any element e of A, e<>0, which divides both a
and b also divides d.

quasi
.

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