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NILRADICAL OF A RING

ABSTRACT

In algebra, the nilradical of a commutative ring is the ideal consisting of the nilpotent elements: Պ_R={f∈R|f^m=0 for some m∈Z_(>0) }. It is thus the radical of the zero ideal. If the nilradical is the zero ideal, the ring is called a reduced ring. The nilradical of a commutative ring is the intersection of all prime ideals. In the non-commutative ring case the same definition does not always work. This has resulted in several radicals generalizing the commutative case in distinct ways; see the article Radical of a ring for more on this. The nilradical of a Lie algebra is similarly defined for Lie algebras.

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