Nilpotents
Nilpotents are elements of a ring in which some positive power equals zero. Specifically, an element x in a ring R is nilpotent if there exists k ≥ 1 with x^k = 0. This notion is most often discussed in rings with identity, where the index of nilpotency is the smallest such k.
In a commutative ring, the collection Nil(R) of all nilpotent elements forms an ideal called the nilradical.
Examples include: in the ring Z/8Z, the elements 2 and 4 are nilpotent since 2^3 = 8 ≡
Several properties hold in commutative rings: the sum of two nilpotent elements is nilpotent, the product of