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The nilradical of a ring

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.

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