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Comaximal Ideals in a Principal Ideal Domain

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,918
Prove that in a prinicpal ideal domain, two ideals (a) and (b) are comaximal if and only if a greatest common divisor of a and b (in which case (a) and (b) are said to be coprine or realtively prime)

Note: (1) Two ideals A and B of the ring R are said to be comaximal if A + B = R

(2) Let I and J be two ideals of R
The sum of I and J is defined as [TEX] I+J = \{ a+b | a \in I, b \in J \} [/TEX]
 

Siron

Active member
Jan 28, 2012
150
Re: Comaximal Ideas in a Principal Ideal Domain

I think you've found the answer on MHF.
 

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,918
Re: Comaximal Ideas in a Principal Ideal Domain

Well ... I am still working on the problem ... but I will be using your guidance regarding the way to progress

At my day job at the moment ... but will use your hint when I return to the problem

Thanks again

Peter
 

jakncoke

Active member
Jan 11, 2013
68
by def, two elements a,b in a PID are relatively prime if there exist, $m_1,m_2 \in $, such
that 1 = $m_1a+m_2b$

now if $a,b$ are relatively prime then

$\{r_1a + r_2b | r_1,r_2 \in R\}$, contains 1, if an ideal contains 1, then that ideal is identical to R.

Now <a> + <b> = $\{g_1a + g_2b | g_1,g_2 \in R\}$
 
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