Discussion
Which of the following is a PID ?
Answer: B
The PID is \(\mathbf{R}(x)[y]\).
\(\mathbf{R}(x)\) is the field of rational functions in \(x\), and a polynomial ring in one variable over a field is a Euclidean domain, hence a principal ideal domain.
The others are not. In \(\mathbf{Z}[x]\) the ideal \((2, x)\) is not principal. In \(\mathbf{C}[x, y]\) the ideal \((x, y)\) is not principal. And \(\mathbf{Z}_{30}\) is not an integral domain at all, since \(2 \cdot 15 = 0\), so it cannot be a PID even though all its ideals are principal.
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