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.

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    $\begingroup$ If $x$ is nilpotent then $1-x$ is a unit. $\endgroup$ – PrimeNumber Jan 26 '11 at 20:47
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    $\begingroup$ @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. $\endgroup$ – Bill Dubuque Jan 30 '11 at 19:16
  • $\begingroup$ The idea in the prior comment is a special case of the method of simpler multiples $\endgroup$ – 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.

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    $\begingroup$ See here for units in the Laurent polynomial ring $R[x,x^{-1}]\ \ $ $\endgroup$ – 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 $\deg(f)$).

  • $\begingroup$ Why does $a_{n-1}b_m+a_nb_{m-1}=0 \Rightarrow (a_n)^2b_{m-1}=0$ ? $\endgroup$ – Nethesis Oct 3 '14 at 0:45
  • $\begingroup$ Wait, got it, though surely that is only true if $a_n \in (Z)R$? $\endgroup$ – Nethesis Oct 3 '14 at 0:46
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    $\begingroup$ @Nethesis If by $Z$ you mean the center, it is probably because the ring is given commutative ;) $\endgroup$ – N. S. Oct 3 '14 at 1:15
  • $\begingroup$ 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. $\endgroup$ – Nethesis Oct 5 '14 at 17:40
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    $\begingroup$ @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 $$(u-v)(u^{n-1}+u^{n-2}v+...+v^{n-1})(u^{-1})^n=1$$ $\endgroup$ – N. S. Apr 20 '15 at 15:47