$R$ has nonzero nilpotent elements...
17926
Zero divisor in $R[x]$
114662
Construction of a specific non-commutative and infinite group (with conditions on the order of the elements)
39049
Ring With Unity As a Direct Sum of Non-Zero Ideals
79957
If $R$ is an infinite ring, then $R$ has either infinitely many zero divisors, or no zero divisors
156443
if $x$ and $y$ commute with $[x y]$, then $[x y]$ is nilpotent
22563
Units and Nilpotents
1973
Traces of all positive powers of a matrix are zero implies it is nilpotent
7034
Prove that $A+I$ is invertible if $A$ is nilpotent
56097
If $A$ and $AB-BA$ commute, show that $AB-BA$ is nilpotent
155451
Upper bound for the rank of a nilpotent matrix , if $A^2 \ne 0$
38346
Commutative Ring: Nilpotent elements closed under addition?
115920
Showing $R$ is a local ring if and only if all elements of $R$ that are not units form an ideal
83115
A set of elements in a reduced unity ring
80616
A ring with few invertible elements
67944
Nilpotent Elements and Intersection of Prime Ideals
1990
Results for elements in Dihedral Groups: $x \notin \langle r\rangle  \Rightarrow rx = xr^{-1}$
28460
Show matrix is nilpotent