If $f$ is a unit in a polynomial ring then $a_0$ is unit and all other coeficients are nilpotent.
35222
Characterizing units in polynomial rings
89313
Maximal ideals of polynomial rings in infinitely many variables
96144
Every element outside the maximal ideal of a local ring is a unit
133046
Showing $[L,rad(L)]$ is nilpotent.
6722
to show $f$ is a polynomial in $z$
69012
How to directly prove every element in a ring with exactly three ideals is either a unit or a zero divisor?
66968
if $a,b$ are nilpotent elements of a commutative ring $R$, show that $a+b$ is also nilpotent
154530
Rings whose elements are partitioned between units and zero-divisors.
158426
Proving lemma about centrality of idempotent elements in a Ring with no nilpotent elements.
156443
if $x$ and $y$ commute with $[x y]$, then $[x y]$ is nilpotent
95146
Polynomial ring $R[X] $ has zero Jacobson radical, for $R$ a domain
77820
Roots in a polynomial ring
130104
Show $4x^2+6x+3$ is a unit in $\mathbb{Z}_8[x]$ (inverting unit + nilpotent)
25748
If $G / Z(G)$ nilpotent then G is nilpotent.
22563
Units and Nilpotents
25009
$\operatorname{tr}(A)=\operatorname{tr}(A^{2})= \ldots = \operatorname{tr}(A^{n})=0$ implies $A$ is nilpotent
111824
Sum of nilpotent and element is a unit in ring?