# Integral Domains and Principal Integral Domains

See attached file.

I am having problems with a couple of abstract algebra problems regarding integral domains and principal integral domains (PIDs).

A commutative ring satisfied the DCCP if <a1> ⊇ <a2> ⊇ <a3> ⊇ ... implies that an ~ an+1 ~ ... for some n ≥ 1.

Show that an integral domain R has DCCP if and only if R is a field.

Let R be a UFD. Show that R is a PID if and only if it satisfies the following condition:

For all a ≠ 0 and b ≠ 0, there exists r and s in R such that gcd(a, b) ~ ra + sb.

https://brainmass.com/math/basic-algebra/integral-domains-principal-integral-domains-428281

#### Solution Preview

Please see the attachment for the complete solution response.

Problem #1

Show that an integral domain (please see the attached file) has DCCP if and only if (please see the attached file) is a field.

Proof: (please see the attached file)

"(please see the attached file)": Since (please see the attached file) is an integral domain, to show that (please see the attached file) is a field, I only need to prove that every non-zero element (please see the attached file) has a multiplicative inverse. Actually, we consider the chain (please see the attached file). Since (please see the attached file) has DCCP, we can find some (please see the attached file), such that (please see the attached file). Then we have (please see the attached file) for some (please see the attached file). Since (please see the attached file) is an ...

#### Solution Summary

Integral domains and principal integral domains are exemplified in this solution.