Principal ideals

• Nov 12th 2009, 11:37 AM
dangkhoa
Principal ideals
Let a, b in R where R is a UFD.
Is it true that (a) intersects (b) = (c) always? Where (a),(b) and(c) are principal ideals
How do you prove it or give a counter-example?

• Nov 12th 2009, 12:53 PM
tonio
Quote:

Originally Posted by dangkhoa
Let a, b in R where R is a UFD.
Is it true that (a) intersects (b) = (c) always? Where (a),(b) and(c) are principal ideals
How do you prove it or give a counter-example?

Well, a UFD is also a PID and since the intersection of two ideals is always an ideal then it MUST be that $(a)\cap (b)=(c)$....

Tonio
• Nov 12th 2009, 02:18 PM
signaldoc
Pid <--> ufd ?
A PID is a UFD, but the converse requires more info. See (for example)

ftp://ftp.cis.upenn.edu/pub/papers/gallier/geomath3.pdf Proposition 5.40

All is not lost, I think. Here is a suggestion: Suppose a and b have common factor k: a=k*a1, b=k*b1. Then (I believe, and invite verification) the intersection of (a) and (b) is (k*a1*b1)=(c). This may require the ideal to be two sided.

Hope this helps....