FEEBLY NIL-CLEAN UNITAL RINGS
0
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
We define and explore feebly nil-clean rings in different aspects. Our results considerably enlarge in certain aspects recent theorems published in J. Algebra and its Applications by various authors quoted in the bibliography list. Specifically, we prove that any feebly nil-clean ring R is decomposed as the direct product of a nil-clean ring and a 3-good ring in which 3 is a nilpotent. In some partial cases, such a ring R can be completely characterized. We also show that the full matrix n×n ring Mn(F) over a field F is feebly nil-clean if, and only if, either F ≅ Z2 or F ≅ Z3.
Abstract
We define and explore feebly nil-clean rings in different aspects. Our results considerably enlarge in certain aspects recent theorems published in J. Algebra and its Applications by various authors quoted in the bibliography list. Specifically, we prove that any feebly nil-clean ring R is decomposed as the direct product of a nil-clean ring and a 3-good ring in which 3 is a nilpotent. In some partial cases, such a ring R can be completely characterized. We also show that the full matrix n×n ring Mn(F) over a field F is feebly nil-clean if, and only if, either F ≅ Z2 or F ≅ Z3.
참고문헌 (15)
- M. Abdolyousefi, "Rings in which elements are sums of tripotents and nilpotents", J. Algebra Appl., 17(3) (2018)
- N. Arora, "Commutative feebly clearingsn", J. Algebra Appl., 16(7) (2017)
- S. Breaz, "Nil-clean matrix rings", Linear Algebra Appl., 439, pp.3115~3119 (2013)[DOI]
- S. Breaz, "Rings in which every element is either a sum or a difference of a nilpotent and an idempotent", J. Algebra Appl., 15(8) (2016)
- H. Chen, "Strongly 2-nil-clean rings", J. Algebra Appl., 16(8) (2017)
- P.V. Danchcv, "Weakly UU rings", Tsukuba J. Math., 40(1), pp.101~118 (2016)[DOI]
- P.V. Danchev, "A lemma on involutions and weakly nil-clean rings", Internat. J. Algebra, 11(2), pp.75~81 (2017)[DOI]
- P.V. Danchev, "On exchange <TEX>${\pi}$<", TEX>-UU unital rings, 2017, pp.39, ~
- A.J. Diesl, "Nil clean rings", J. Algebra, 383, pp.197~211 (2013)[DOI]
- Y. Hirano, "Rings in which every element is the sum of two idempotents", Bull. Austral. Math. Soc., 37, pp.161~164 (1988)[DOI]
- T.Y. Lam, "A First Course in Noncommutative Rings, Second Edition, Graduate Texts in Math., Vol. 131" (2001), Springer-Verlag
- W.K. Nicholson, "Lifting idempotents and exchange Tings", Trans. Amer. Math. Soc., 229, pp.269~278 (1977)[DOI]
- J. Ster, "Rings in which nilpotentsform a subring", Carpath. J. Math., 32(2), pp.251~258 (2016)
- J. Ster, "Nil-clean quadratic elements", J. Algebra Appl., 16(8) (2017)
- Y. Zhou, "Rings in which elements are sums of nilpotents, idempotents and tripotents", J. Algebra Appl., 17 (2018)