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An element of a ring
is called nilpotent if there exists some positive integer n such that
. ↑
The term goes back to Benjamin Peirce and was motivated by elements of an algebra that vanish when raised to a power.
The set of all nilpotent elements of a commutative ring with identity is called the nilradical of
, and it is usually denoted by
.
The nilradical of R is the radical of the zero ideal . ↑
Theorem. If is the niradical of a ring
, then the factor ring
has no non-zero nilpotent elements (that is, it is a reduced ring ↑)
Theorem. The nilradical of a ring is the intersection of all prime ideals of
.
For non-commutative rings, there are several analogues of the nilradical.
Cite this web-page as:
Štefan Porubský: Nilradical.