This definition can be applied in particular to square matrices. The matrix

In the factor ring **Z**/9**Z**, the class of 3 is nilpotent because 3^{2} is congruent to 0 modulo 9.

No nilpotent element can be a unit (except in the trivial ring {0} which has only a single element 0=1). All non-zero nilpotent elements are zero divisors.

An *n*-by-*n* matrix *A* with entries from a field is nilpotent if and only if its characteristic polynomial is *T*^{n}, which is the case if and only if *A*^{n} = 0.

The nilpotent elements from a commutative ring form an ideal; this is a consequence of the binomial theorem. This ideal is the nilradical of the ring. Every nilpotent element in a commutative ring is contained in every prime ideal of that ring, and in fact the intersection of all these prime ideals is equal to the nilradical.

If *x* is nilpotent, then 1-*x* is a unit, because *x*^{n} = 0 entails

- (1-
*x*) (1 +*x*+*x*^{2}+ ... +*x*^{n-1}) = 1 -*x*^{n}= 1.