Research Article
Year 2025,
, 807 - 821, 24.06.2025
Abstract
This article studies the ring structure arising from products of idempotents and nilpotents. Thus the argument is concerned essentially with the one-sided IQNN property of rings. We first prove that if the $2$ by $2$ full matrix ring over a principal ideal domain $F$ of characteristic zero is right IQNN then $F$ contains infinitely many non-integer rational numbers; and that the concepts of right IQNN and right quasi-Abelian are independent of each other. We next introduce a ring property, called right IAN, as a generalization of both right IQNN and right quasi-Abelian; and provide several kinds of methods to construct right IAN rings. In the procedure, we also show that the right IQNN and right IAN do not go up to polynomial rings.
References
- [1] H. Chen, Exchange rings with artinian primitive factors, Algebra Represent. Theory. 2, 201–207, 1999.
- [2] E.-K. Cho, T.K. Kwak, Y. Lee, Z. Piao and Y. Seo, A structure of noncentral idempotents, Bull. Korean Math. Soc. 55, 25–40, 2018.
- [3] A.J. Diesl, Nil clean rings, J. Algebra 383, 197–211, 2013.
- [4] J.L. Dorroh, Concerning adjunctions to algebras, Bull. Amer. Math. Soc. 38, 85–88, 1932.
- [5] G. Ehrlich, Unit-regular rings, Portugal Math. 27, 209–212, 1968.
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- [7] K.R. Goodearl, Von Neumann Regular Rings, London/UK, Pitman Publishing Limited, 1979.
- [8] J. Huang, T.K. Kwak, Y. Lee and Z. Piao, Structure of idempotents in polynomial rings and matrix rings, Bull. Korean Math. Soc. 60, 1321–1334, 2023.
- [9] C. Huh, H.K. Kim and Y. Lee, p.p. rings and generalized p.p. rings, J. Pure Appl. Algebra 167, 37–52, 2002.
- [10] N.K. Kim and Y. Lee, Armendariz rings and reduced rings, J. Algebra 223, 477–488, 2000.
- [11] T.K. Kwak, S.I. Lee and Y. Lee, Quasi-normality of idempotents on nilpotents, Hacet. J. Math. Stat. 48, 1744–1760, 2019.
- [12] J. Lambek, Lectures on Rings and Modules, Waltham/USA, Blaisdell Publishing Company, 1966.
- [13] W.K. Nicholson and Y. Zhou, Clean general rings, J. Algebra 291, 297–311, 2005.
Year 2025,
, 807 - 821, 24.06.2025
Abstract
References
- [1] H. Chen, Exchange rings with artinian primitive factors, Algebra Represent. Theory. 2, 201–207, 1999.
- [2] E.-K. Cho, T.K. Kwak, Y. Lee, Z. Piao and Y. Seo, A structure of noncentral idempotents, Bull. Korean Math. Soc. 55, 25–40, 2018.
- [3] A.J. Diesl, Nil clean rings, J. Algebra 383, 197–211, 2013.
- [4] J.L. Dorroh, Concerning adjunctions to algebras, Bull. Amer. Math. Soc. 38, 85–88, 1932.
- [5] G. Ehrlich, Unit-regular rings, Portugal Math. 27, 209–212, 1968.
- [6] K.E. Eldridge, Orders for finite noncommutative rings with unity, Amer. Math. Monthly 75, 512–514, 1966.
- [7] K.R. Goodearl, Von Neumann Regular Rings, London/UK, Pitman Publishing Limited, 1979.
- [8] J. Huang, T.K. Kwak, Y. Lee and Z. Piao, Structure of idempotents in polynomial rings and matrix rings, Bull. Korean Math. Soc. 60, 1321–1334, 2023.
- [9] C. Huh, H.K. Kim and Y. Lee, p.p. rings and generalized p.p. rings, J. Pure Appl. Algebra 167, 37–52, 2002.
- [10] N.K. Kim and Y. Lee, Armendariz rings and reduced rings, J. Algebra 223, 477–488, 2000.
- [11] T.K. Kwak, S.I. Lee and Y. Lee, Quasi-normality of idempotents on nilpotents, Hacet. J. Math. Stat. 48, 1744–1760, 2019.
- [12] J. Lambek, Lectures on Rings and Modules, Waltham/USA, Blaisdell Publishing Company, 1966.
- [13] W.K. Nicholson and Y. Zhou, Clean general rings, J. Algebra 291, 297–311, 2005.
There are 13 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Pure Mathematics (Other) |
| Journal Section | Mathematics |
| Authors | |
| Early Pub Date | August 27, 2024 |
| Publication Date | June 24, 2025 |
| Submission Date | January 4, 2024 |
| Acceptance Date | June 14, 2024 |
| Published in Issue | Year 2025 |