# Characterizing units in polynomial rings

I am trying to prove a result, for which I have got one part, but I am not able to get the converse part.

Theorem. Let $R$ be a commutative ring with $1$. Then $f(X)=a_{0}+a_{1}X+a_{2}X^{2} + \cdots + a_{n}X^{n}$ is a unit in $R[X]$ if and only if $a_{0}$ is a unit in $R$ and $a_{1},a_{2},\dots,a_{n}$ are all nilpotent in $R$.

Proof. Suppose $f(X)=a_{0}+a_{1}X+\cdots +a_{n}X^{n}$ is such that $a_{0}$ is a unit in $R$ and $a_{1},a_{2}, \dots,a_{r}$ are all nilpotent in $R$. Since $R$ is commutative, we get that $a_{1}X,a_{2}X^{2},\cdots,a_{n}X^{n}$ are all nilpotent and hence also their sum is nilpotent. Let $z = \sum a_{i}X^{i}$ then $a_{0}^{-1}z$ is nilpotent and so $1+a_{0}^{-1}z$ is a unit. Thus $f(X)=a_{0}+z=a_{0} \cdot (1+a_{0}^{-1}z)$ is a unit since product of two units in $R[X]$ is a unit.

I have not been able to get the converse part and would like to see the proof for the converse part.

• If $x$ is nilpotent then $1-x$ is a unit. – PrimeNumber Jan 26 '11 at 20:47
• @Chandru1: $\ u$ unit, $z^n = 0\ \Rightarrow\ u-z\ |\ u^n - z^n = u^n,\$ so $\ u - z\$ is a unit, being a divisor of the unit $u^n\:.\$ Thus $\$ unit + nilpotent = unit. – Bill Dubuque Jan 30 '11 at 19:16
• The idea in the prior comment is a special case of the method of simpler multiples – Bill Dubuque Feb 21 at 8:54

If $$R$$ is a domain then easily $$f(X)$$ a unit implies that $$a_i = 0$$ for $$i>0$$. Now $$R\to R/\mathfrak p$$, for $$\mathfrak p$$ prime, reduces to the domain case, yielding that the $$a_i$$, $$i>0$$ are in every prime ideal. But the intersection of all prime ideals is the nilradical, the set of all nilpotent elements - as you proved a few days ago.

Remark  This is a prototypical example of reduction to domains by factoring out prime ideals.

• See here for units in the Laurent polynomial ring $R[x,x^{-1}]\ \$ – Bill Dubuque Jun 28 '19 at 12:43

Let $$f=\sum_{k=0}^n a_kX^k$$ and $$g= \sum_{k=0}^m b_kX^k$$. If $$f g=1$$, then clearly $$a_0,b_0$$ are units and:

$$a_nb_m=0$$ $$a_{n-1}b_m+a_nb_{m-1}=0 \Rightarrow (a_n)^2b_{m-1}=0$$ $$a_{n-2}b_m+a_{n-1}b_{m-1}+a_nb_{m-2}=0 \Rightarrow (a_n)^3b_{m-2}=0$$ $$.....$$ $$.....+a_{n-2}b_2+a_{n-1}b_1+a_nb_0=0 \Rightarrow (a_n)^{m+1}b_{0}=0$$

Since $$b_0$$ is an unit, it follows that $$(a_n)^{m+1}=0$$.

Hence, we proved that $$a_n$$ is nilpotent, but this is enough. Indeed, since $$f$$ is invertible, $$a_nx^n$$ being nilpotent implies that $$f-a_nX^n$$ is unit and we can repeat (or more rigorously, perform induction on $$\deg(f)$$).

• Why does $a_{n-1}b_m+a_nb_{m-1}=0 \Rightarrow (a_n)^2b_{m-1}=0$ ? – Nethesis Oct 3 '14 at 0:45
• Wait, got it, though surely that is only true if $a_n \in (Z)R$? – Nethesis Oct 3 '14 at 0:46
• @Nethesis If by $Z$ you mean the center, it is probably because the ring is given commutative ;) – N. S. Oct 3 '14 at 1:15
• Ah right, sorry, I was looking for an answer about a non commutative ring and din't see that in the question, my bad @N.S. – Nethesis Oct 5 '14 at 17:40
• @zed111 In a commutative ring you have unit-nilpotent=unit. The proof is easy: if u is unit and v is nilpotent, then $v^n=0$ which means $$u^n=u^n-v^n=(u-v)(u^{n-1}+u^{n-2}v+...+v^{n-1})$$ This shows that $$(u-v)(u^{n-1}+u^{n-2}v+...+v^{n-1})(u^{-1})^n=1$$ – N. S. Apr 20 '15 at 15:47