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1985•Proceedings of the American Mathematical SocietyOpen access

Nil Derivations and Chain Conditions in Prime Rings

L. O. Chung, Y. Kobayashi

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Abstract

It is known that in a prime ring, a derivation nilpotent on a nonzero ideal must also be nilpotent in the entire ring. In this paper we show that a derivation of a prime ring is not necessarily nil even though it is nil on a nonzero ideal. It is nil if the ring satisfies the ascending chain condition on left (right) annihilator ideals.

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What this paper is about

It is known that in a prime ring, a derivation nilpotent on a nonzero ideal must also be nilpotent in the entire ring. In this paper we show that a derivation of a prime ring is not necessarily nil even though it is nil on a nonzero ideal. It is nil if the ring satisfies the ascending chain condition on left (right) annihilator ideals.

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Available abstract

It is known that in a prime ring, a derivation nilpotent on a nonzero ideal must also be nilpotent in the entire ring. In this paper we show that a derivation of a prime ring is not necessarily nil even though it is nil on a nonzero ideal. It is nil if the ring satisfies the ascending chain condition on left (right) annihilator ideals.

Key concepts: Associated prime, Annihilator, Nilpotent, Mathematics, Radical of a ring, Prime ring, Ideal (ethics), Prime (order theory)

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