Discussion
Let I be the ideal in the polynomial ring \(\mathbf{Z}_{4}[X]\) generated by the polynomial 1 + 2X. Then :
Answer: D
I is the unit ideal, the whole ring.
In \(\mathbf{Z}_{4}[X]\), \(1 + 2X\) is a unit:
\[ (1 + 2X)^{2} = 1 + 4X + 4X^{2} = 1 \]
because \(4 = 0\) in \(\mathbf{Z}_{4}\). An ideal containing a unit contains \(1\) and therefore everything, so \(I = \mathbf{Z}_{4}[X]\).
A prime or maximal ideal must be proper by definition, so A and B fail, and C, 'proper but not prime', fails because I is not proper at all. The trap is that a polynomial of degree 1 cannot be a unit over a field or an integral domain, but \(\mathbf{Z}_{4}\) has the nilpotent element 2.
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