Nilpotent

A nilpotent element is a term used in ring theory, a branch of mathematics. An element of a ring is called nilpotent if it repeatedly multiplied by itself gives zero.

Definition

An element of a ring is nilpotent when there is a positive integer, so that is valid. An ideal is known as nilpotent when a positive natural number exists, so that is valid.

Examples

• The residue class ring the residue classes of 0, 2, 4 and 6 are nilpotent, since each have their third power is congruent to 0 modulo 8. In this ring, each element is either nilpotent or a unit.

• The residue class ring the nilpotent elements are exactly the cosets of 0 and 6

• The zero element of a ring is always nilpotent, there is.

Properties

The set of all nilpotent elements of a commutative ring is an ideal, the so-called nilradical.

The average of all prime ideals in a commutative ring with 1 is exactly the nilradical.

Is in the following with a ring member of a nilpotent and the smallest natural number.

• Is, then, and is a zero divisor, then, and.

If, in addition, a ring with 1 and not the zero ring, then:

• Is not invertible (with respect to the multiplication), because of a ring element follows the Opposition ( elected was minimal ).
• Is invertible, because it is.
• If a unit of which commutes with, then is also invertible, what one sees than by consideration of the representation.

Be a residue class ring and the product of all prime divisors of, that all primes occurring in the prime factorization of. For example, for is. Then the nilpotent elements of exactly the residue classes of integers that are multiples of are. The idea of ​​the proof is the following: If the largest exponent that occurs in the prime factorization of, then is a multiple of; each number for which a power is a multiple of, already must themselves have every prime divisor of.

A ring which does not contain elements other than the zero nilpotent is called reduced.