General Definition of Relatively prime

[snoble]

I am wondering about the general definition of relatively prime in terms of commutative rings.

Specifically if I have my first definition being that given a commutative ring R if r_1 and r_2 are relatively prime then if $$r_1 k\in r_2R$$ then $$k \in r_2 R$$. And vice versa. In other words if $$r_2 | kr_1$$ then $$r_2 | k$$.

My second definition is if r_1 and r_2 are relatively prime then $$\exists x,y\in R$$ such that $$xr_1 +yr_2 = 1$$ (yes I'm assuming all rings have a unit)

So I'm wondering for what types of commutative rings are these two definitions equivalent (I'm guessing always or almost always) and where can I find a proof of that. Notice the Euclidean algorithm depends on an ordering which you may not have here (at least as I know the algorithm).

Another way to think of the problem is how do I show given the first definition that <r_1> and <r_2> are comaximal: ie $$<r_1>+<r_2>=R$$. This is the actual problem I've been thinking about.

Thanks,
Steven

 

[mathwonk]

you second definition seems to be sort of "relatively maximal" as opposed to just relatively prime.

for example in a polynomial ring k[X,Y,Z,], then X and Y are relatively prime in the first sense but not the second.

but in a smaller ring like a pid, say k[X], the first definition is true.

for example in a domain, a prime element is one that generates a prime ideal, i.e. x is prime if whenever yz is divisible by x then ether y or z is.

then in a unique factorization domain, ike any polynomial ring in any number if variables, two elements are relatively prime in your first sense if they have no common prime factors.

but the second sense is still not true for them if the ring is large like a polynomial ring in two or more variables. i.e. you are saying there that the only way two elements are relatively prime is if they generate the nuit ideal. so if there are some large non unit ideas out there like (X,Y), (X,Y,Z), (X,Y,Z,W),... this is not the case.

of course your second property immediately implies the first one. do you see how to prove that? the proof uses what I call the "three term principle" in my gentle introductions to proofs. (if x divides two of three terms in an equation, it divides also the third.)

 

[matt grime]

In general, from what I can recall from many sci.maths posts to james harris, elements are coprime if 1 is in the ideal they generate, ie (x,y) =1, if there is a linear combination of x and y sch that ax+by=1

 

[mathwonk]

that may be current terminology, but it follows from that definition that there are no coprime pairs of (non unit) elements in most rings of krull dimension at least two, for example k[X,Y].

that property is standard for pid's of course.

That property is called "comaximal" in Commutative Algebra, vol I, by Zariski Samuel, page 176.

I do not know James Harris, but I was actually unable to find either of the terms "coprime" or "relatively prime" listed in the indices of any of my commutative algebra texts, Zariski Samuel, Atiyah MacDonald, Matsumura, Reid, or Eisenbud.

a search on "james harris, mathematics" turned up as series of articles about "kooks, cranks, and loons". Someone of that name seems bound to publish a record number of false elementary "proofs" of fermat's last theorem.

presumably we are relying on someone else for this definition, but even that james harris might know this kind of thing.

 

[Hurkyl]

Atiyah-MacDonald talks about "coprime" on page 7, and it's synonymous with "comaximal". However, these terms were applied to ideals, not individual elements.

Aha, I finally found my copy of Jacobson. (I really need to organize better. ) It defines that a and b are relatively prime iff gcd(a, b) = 1, and goes on to say that happens only if a or b is a unit, or that no irreducible element divides both a and b.

(It's in a section on Factorial Monoids and Rings)

 

[snoble]

Is it fair to say that the general consensus is to take the second definition as the standard which happens to imply the first? But as in mathwonk's example the first doesn't imply the second?

Oh by the by (out of curiousity) how does Jacobson define gcd on a ring without order? I'm sure I'll kick myself when I hear it :tongue:

 

[mathwonk]

of course it is clear that hurkyl's citation of the definition of relatively prime in Jacobson as saying that gdc(a,b) = 1, is not the second definition but the first one.

the definition in atiyah macdonald is on the other hand the first definition.

i certainly hope there is no consensus that the accepted definition is the second one as that flies in the face of the meanings of the words for all uses i know of, except for pid's like the integers.

indeed i am puzzled that atiyah macdonald sanctioned that term for the definition of comaximal. Of course atiyah, fields medalist or not, is a topologist, not an algebraist.

 

[mathwonk]

what do, you mean by order? oh i see what you are thinking; you are thinking "g" in gcd means "greatest"; observe that gcd(a,b) means not the "greatest"common divisor, but rather c is the gcd of a and b if and only if every common divisor of a and b also divides c. I.e. gcd really means the "universal common divisor".

they do not alweays exist of course, but do exist in any unique factorization domain.

 

[Hurkyl]

I'd just like to reiterate that Atiyah-Macdonald applied the terms to ideals, not to individual elements.

 

[mathwonk]

that is so. but for individual elements that would mean principal ideals, and there it seems poorly applicable.

or maybe their point is to distinguish the term "coprime" from "relatively prime".

 

[matt grime]

Oh, the james harris thing. He is a crank of the finest order, and one of his claims was about things that are "coprime" in some ring. He didn't define it, and is completely wrong, but a lot of mathematicians chipped in and gave their definition of what it meant to be coprime.

 

[mathwonk]

that stuff keeps us entertained. james harris seems one of those cranks that people are rather fond of as opposed to being annoyed at.