Assignment:
A commutative ring satisfied the DCCP if 1> ⊇ 2> ⊇ 3> ⊇ ... 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.
Provide complete and step by step solution for the question and show calculations and use formulas.