nilradical
Some elements of a ring are, in a sense, infinitesimally small: raise them to a high enough power and they collapse to zero. Such nilpotent elements are the algebraic ghosts of geometry — they record 'thickening' or fuzz that a clean geometric picture would not see. The nilradical bundles all these ghosts together into a single ideal, measuring exactly how far a ring is from being free of such fuzz.
For a commutative ring R, the nilradical Nil(R) is the set of all nilpotent elements: those a with a^n = 0 for some positive integer n. This set is an ideal, and a beautiful fact is that it equals the intersection of all prime ideals of R. A ring is called reduced when its nilradical is zero, i.e. it has no nonzero nilpotents. The quotient R/Nil(R) is always reduced; geometrically, passing to it forgets the infinitesimal thickening and remembers only the underlying point set.
The identity Nil(R) = intersection of all primes is the prime-avoidance heart of the matter: a is nilpotent if and only if a lies in every prime ideal. Beware the contrast with the Jacobson radical, the intersection of all MAXIMAL ideals — the two coincide for many rings but differ in general, and the nilradical is always contained in the Jacobson radical.
In Z/8Z the nilpotent elements are 0, 2, 4, 6 (since 2^3 = 8 = 0), so the nilradical is (2) = {0, 2, 4, 6}. In Z/12Z, since 6^2 = 36 = 0, the nilradical is (6) = {0, 6}.
Nilradicals of two small finite rings.
The nilradical is the radical of the zero ideal: Nil(R) = sqrt((0)). More generally the radical of any ideal I is the preimage of Nil(R/I).