Find single generators for the ideals \((85,1+13i)\) and \((47-13i, 53+56i)\) in \(\mathbb{Z}[i]\).

Prove that the quotient ring \(\mathbb{Z}[i]/I\) is finite for any nonzero ideal \(I\) of \(\mathbb{Z}[i]\).

Prove that in a PID two ideals \((a)\) and \((b)\) are comaximal if and only if the gcd of \(a\) and \(b\) is \(1\).

An integeral domain \(R\) in which every ideal generated by two elements is principal is called a Bezout domain.

1. Prove that the integral domain \(R\) is a Beztou Domain if and only if every pair of elements \(a,b\) of \(R\) has a gcd \(d\) in \(R\) that can be written as an \(R\)-linear combination of \(a\) and \(b\).

2. Prove that every finitely generated ideal of a Bezout Domain is principal.

3. Let \(F\) be the fraction field of the Bezout Domain \(R\). Prove that every element of \(F\) can be written in the form \(a/b\) with \(a,b \in R\) and \(a\) and \(b\) relative prime.

Prove that if \(R\) is a PID and \(D\) is a multiplicatively closed subset of \(R\), then \(D^{-1}R\) is also a PID.

Prove that \(R\) is a PID if and only if \(R\) is a UFD that is also a Bezout Domain.