On $ S $-principal right ideal rings

Jongwook Baeck; Jongwook Baeck

doi:10.3934/math.2022673

AIMS Mathematics

2022, Volume 7, Issue 7: 12106-12122. doi: 10.3934/math.2022673

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Research article

On $ S $-principal right ideal rings

Jongwook Baeck ^,

School of Basic Sciences, Hanbat National University, Daejeon 34158, Republic of Korea
Correction on: AIMS Mathematics 8: 12694–12695

Received: 11 March 2022 Revised: 15 April 2022 Accepted: 20 April 2022 Published: 22 April 2022
MSC : 16P99, 16D99, 16D25, 16P40

Let $ S $ be a multiplicative subset of a ring $ R $. A right ideal $ A $ of $ R $ is referred to as $ S $-principal if there exist an element $ s \in S $ and a principal right ideal $ aR $ of $ R $ such that $ As \subseteq aR \subseteq A $. A ring is referred to as an $ S $-principal right ideal ring ($ S $-PRIR) if every right ideal is $ S $-principal. This paper examines $ S $-PRIRs, which extend the notion of principal right ideal rings. Various examples, including several extensions of $ S $-PRIRs are investigated, and some practical results are proven. A noncommutative $ S $-PRIR that is not a principal right ideal ring is found, and the $ S $-variants of the Eakin-Nagata-Eisenbud theorem and Cohen's theorem for $ S $-PRIRs are proven.
- $ S $-principal right ideal ring,
- $ S $-principal right ideal domain,
- right $ S $-Noetherian ring,
- $ S $-principal right ideal,
- Eakin-Nagata-Eisenbud theorem,
- Cohen's theorem
Citation: Jongwook Baeck. On $ S $-principal right ideal rings[J]. AIMS Mathematics, 2022, 7(7): 12106-12122. doi: 10.3934/math.2022673

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Abstract

Let $ S $ be a multiplicative subset of a ring $ R $. A right ideal $ A $ of $ R $ is referred to as $ S $-principal if there exist an element $ s \in S $ and a principal right ideal $ aR $ of $ R $ such that $ As \subseteq aR \subseteq A $. A ring is referred to as an $ S $-principal right ideal ring ($ S $-PRIR) if every right ideal is $ S $-principal. This paper examines $ S $-PRIRs, which extend the notion of principal right ideal rings. Various examples, including several extensions of $ S $-PRIRs are investigated, and some practical results are proven. A noncommutative $ S $-PRIR that is not a principal right ideal ring is found, and the $ S $-variants of the Eakin-Nagata-Eisenbud theorem and Cohen's theorem for $ S $-PRIRs are proven.

References

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