$f$ is a unit in $R$ if and only if $a_0$ is a unit and $a_n, ..., a_1$ are nilpotent
12596
Show that $f(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n\in R[x]$ is nilpotent iff $a_0,a_1,a_2,\ldots,a_n$ are nilpotent
69869
if $f$ is continuous on $\mathbb{R}$ and $f(r)=0,r \in \mathbb{Q}$, then $f(x)=0,x \in \mathbb{R}$
111129
Show that $[K:F]_s = [K:L]_s [L:F]_s$ and $[K:F]_i = [K:L]_i [L:F]_i$.
64484
Left ideal $I$ is a direct summand of $R$ iff $I = Rr$, $r^2 = r$
125299
If A+tB is nilpotent for n+1 distinct values of t, then A and B are nilpotent.
35222
Characterizing units in polynomial rings
37867
If $0$ is the only eigenvalue of a linear operator, is the operator nilpotent
84510
If $f(\mathbb{C})\subset \mathbb{C}-[0,1]$ then $f$ is constant
464
Prove that the ideal $(X_1-a_1,...,X_n-a_n)$ is maximal in $K[X_1,\dots,X_n]$
67291
Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least n+1 zeros on $(-1,1)$
103830
Each element of a ring is either a unit or a nilpotent element iff the ring has a unique prime ideal
138632
Proof that if $a > 0$, then $\frac{1}{a} > 0$ using just field and order axioms
39788
$a+1,a-1$ invertible for nilpotent element
25009
$\operatorname{tr}(A)=\operatorname{tr}(A^{2})= \ldots = \operatorname{tr}(A^{n})=0$ implies $A$ is nilpotent
45310
If $H\leq Z (G)$ and $G/H$ is nilpotent, then $G$ is nilpotent.
63171
Translate of an ideal by $a$ coincides with the ideal iff  $a \in I$
133019
Nilpotent elements in $\mathbb Z_n$