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1971•Canadian Mathematical BulletinOpen access

On Prime Rings with Ascending Chain Condition on Annihilator Right Ideals and Nonzero Infective Right Ideals

Kwangil Koh, A. C. Mewborn

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Abstract

If I is a right ideal of a ring R, I is said to be an annihilator right ideal provided that there is a subset S in R such that I is said to be injective if it is injective as a submodule of the right regular R-module RR. The purpose of this note is to prove that a prime ring R (not necessarily with 1) which satisfies the ascending chain condition on annihilator right ideals is a simple ring with descending chain condition on one sided ideals if R contains a nonzero right ideal which is injective.

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If I is a right ideal of a ring R, I is said to be an annihilator right ideal provided that there is a subset S in R such that I is said to be injective if it is injective as a submodule of the right regular R-module RR. The purpose of this note is to prove that a prime ring R (not necessarily with 1) which satisfies the ascending chain condition on annihilator right ideals is a simple ring with descending chain condition on one sided ideals if R contains a nonzero right ideal which is injective.

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Available abstract

If I is a right ideal of a ring R, I is said to be an annihilator right ideal provided that there is a subset S in R such that I is said to be injective if it is injective as a submodule of the right regular R-module RR. The purpose of this note is to prove that a prime ring R (not necessarily with 1) which satisfies the ascending chain condition on annihilator right ideals is a simple ring with descending chain condition on one sided ideals if R contains a nonzero right ideal which is injective.

Key concepts: Annihilator, Mathematics, Associated prime, Minimal ideal, Ideal (ethics), Radical of a ring, Ring (chemistry), Semiprime ring

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