Comaximal Ideas in a Principal Ideal Domain

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

Re: Comaximal Ideas in a Principal Ideal Domain

Re: Comaximal Ideas in a Principal Ideal Domain

Thanks for the help

One immediate problem for me is that I cannot see why the following is true:

Can you please show me explicitly why this is the case.

Thanks

Peter

Re: Comaximal Ideas in a Principal Ideal Domain

Prove and

Re: Comaximal Ideals in a Principal Ideal Domain

Can clearly see that proving and is the way to go ... but ...

How do we even know that R has a 1?

Can someone help?

Peter

Re: Comaximal Ideals in a Principal Ideal Domain

I think I just answered my most recent post.

Just checked definitions of PID and Integral domain

A PID is an integral domain in which every ideal is principal

and

An integral domain is a commutative ring **with identity **having no zero divisors

So by definition a PID has a 1!

Can someone confirm this is correct?

Peter

Re: Comaximal Ideals in a Principal Ideal Domain

1 just denotes the multiplicative identity on the ring.

EDIT: Yes, you are right. A PID has to contain a multiplicative identity.