Discussion
Which of the following is a principal ideal of \(\mathbf{Z}[x]\) ?
Answer: D
The principal ideal is \((2x, 3x, x^{2})\).
Since \(3x - 2x = x\), the ideal contains \(x\); and each generator \(2x, 3x, x^{2}\) is a multiple of \(x\). So \((2x, 3x, x^{2}) = (x)\), which is principal.
The others are not principal. \((2, x)\) consists of polynomials with even constant term; a single generator would have to divide both 2 and \(x\), so it would be \(\pm 1\), but 1 is not in the ideal. The same argument works for \((3, x)\). And \((x, x^{2} - x, x + 4) = (x, 4)\), which fails for the same reason.
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