Notation of ideals in ring theory

  • #1

Main Question or Discussion Point

So right now i'm trying to solve this problem in ring theory for homework. The question pertains to proving that in a PID, D, if a,b are elements of D then the gcd of a and b can be written as a linear combination. In any event I know where I have to go but I'm stuck on this one bit of notation.

<a,b> = <d>.

I've never seen notation for an ideal like that with 2 elements separated by a comma. I'd appreciate any insight into this.

Oh and here's the site from wolfram where I originally discovered it:
http://mathworld.wolfram.com/PrincipalIdealDomain.html
 

Answers and Replies

  • #2
370
0
<a,b> is any linear combination of a and b.

That is, <a,b>={as+bt|s,t are in R}
 
  • #3
Thanks man. You rock my socks!
 
  • #4
Okay well, it turned out that the idea from wolfram was more confusing than helpful. There's a scarce amount of information in my text and notes on PID's. The only conjectures that I feel safe in making: if given gcd(a,b)=d then gcd(a,d)=d and gcd(b,d)=d. Not sure how helpful those are though. I think what I'm most confused about is Wolfram's assertion that <a,b> is an ideal for any a,b. I can't verify this anywhere.
 
  • #5
Nevermind, I figured it out. Thanks for the insight Ziox!
 

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