Let \(R\) be a commutative ring with no nonzero nilpotent elements. Show that the only units in the polynomial ring \(R[x]\) are the units of \(R\), regarded as constant polynomials.
Find all units in the polynomial ring \(\mathbb Z_4[x]\).
Solution
1. Let\[ f(x)=a_0+a_1x+\cdots+a_nx^n\in R[x] \]be a unit, with inverse \(g(x)\). Then every coefficient \(a_i\) for \(i>0\) is nilpotent.
Let \(\mathfrak p\) be any prime ideal of \(R\). Reducing the identity \(fg=1\) modulo \(\mathfrak p\) gives a unit \[ \overline f\in (R/\mathfrak p)[x]. \] Since \(R/\mathfrak p\) is an integral domain, degrees add for nonzero polynomials. Hence the only units in \((R/\mathfrak p)[x]\) are nonzero constants. Therefore \[ a_i\in\mathfrak p \qquad(i>0) \] for every prime ideal \(\mathfrak p\). Thus each \(a_i\) belongs to the intersection of all prime ideals, which is the nilradical of \(R\); equivalently, each \(a_i\) is nilpotent.
2. If \(R\) has no nonzero nilpotent elements, then every unit of \(R[x]\) is a constant unit of \(R\).
By 1, all coefficients of positive degree vanish. Hence \(f=a_0\in R\). From \(fg=1\), comparison of constant terms gives \(a_0b_0=1\), so \(a_0\in R^\times\). Conversely every unit of \(R\) is plainly a unit in \(R[x]\).
3. A polynomial\[ f(x)=a_0+a_1x+\cdots+a_nx^n\in\mathbb Z_4[x] \]is a unit only if \(a_0\) is odd and every \(a_i\) for \(i>0\) is even.
Reduce modulo \(2\). The image of a unit of \(\mathbb Z_4[x]\) is a unit of \(\mathbb F_2[x]\), hence must equal the nonzero constant \(1\). Therefore \[ a_0\equiv1\pmod2, \qquad a_i\equiv0\pmod2\quad(i>0). \]
4. Conversely, every polynomial with odd constant coefficient and even positive-degree coefficients is a unit in \(\mathbb Z_4[x]\).
Write \[ f(x)=u+n(x), \] where \(u=a_0\in\{1,3\}\) is a unit and every coefficient of \(n(x)\) is divisible by \(2\), with zero constant term. Then \[ n(x)^2=0 \] because every coefficient of the square is divisible by \(4\). Hence \[ f=u\bigl(1+u^{-1}n(x)\bigr) \] has inverse \[ f^{-1}=u^{-1}\bigl(1-u^{-1}n(x)\bigr), \] since \((u^{-1}n(x))^2=0\).
5. Therefore the units of \(\mathbb Z_4[x]\) are exactly\[ \boxed{ \left\{a_0+a_1x+\cdots+a_nx^n: a_0\in\{1,3\},\ a_i\in\{0,2\}\ (i>0) \right\}. } \]