On reversibility related to idempotents
2
조회
0
다운로드
2
인용
0
참고문헌
키워드
초록
This article concerns a ring property which preserves the reversibility of elements at nonzero idempotents. A ring $R$ shall be said to be {\it quasi-reversible} if $0\neq ab\in I(R)$ for $a, b\in R$ implies $ba\in I(R)$, where $I(R)$ is the set of all idempotents in $R$. We investigate the quasi-reversibility of $2$ by $2$ full and upper triangular matrix rings over various kinds of reversible rings, concluding that the quasi-reversibility is a proper generalization of the reversibility. It is shown that the quasi-reversibility does not pass to polynomial rings. The structure of Abelian rings is also observed in relation with reversibility and quasi-reversibility.
Abstract
This article concerns a ring property which preserves the reversibility of elements at nonzero idempotents. A ring $R$ shall be said to be {\it quasi-reversible} if $0\neq ab\in I(R)$ for $a, b\in R$ implies $ba\in I(R)$, where $I(R)$ is the set of all idempotents in $R$. We investigate the quasi-reversibility of $2$ by $2$ full and upper triangular matrix rings over various kinds of reversible rings, concluding that the quasi-reversibility is a proper generalization of the reversibility. It is shown that the quasi-reversibility does not pass to polynomial rings. The structure of Abelian rings is also observed in relation with reversibility and quasi-reversibility.
참고문헌 (16)
- D.D. Anderson, "Semigroups and rings whose zero products commute", Communications in Algebra, 27(6), pp.2847~2852 (1999), Informa UK Limited[DOI]
- Ayman Badawi, "On abelian π-regular rings", Communications in Algebra, 25(4), pp.1009~1021 (2007), Informa UK Limited[DOI]
- Howard E. Bell, "Near-rings in which each element is a power of itself", Bulletin of the Australian Mathematical Society, 2(3), pp.363~368 (1970), Cambridge University Press (CUP)[DOI]
- P. M. Cohn, "Reversible Rings", Bulletin of the London Mathematical Society, 31(6), pp.641~648 (1999), Wiley[DOI]
- K. R. Goodearl, "von Neumann Regular Rings" (1979), Pitman (Advanced Publishing Program)
- Chan Huh, "p.p. rings and generalized p.p. rings", Journal of Pure and Applied Algebra, 167(1), pp.37~52 (2002), Elsevier BV[DOI]
- Seo Un Hwang, "Structure and topological conditions of NI rings", Journal of Algebra, 302(1), pp.186~199 (2006), Elsevier BV[DOI]
- 정다운, "On properties related to reversible rings", 대한수학회보, 52(1), pp.247~261 (2015), 대한수학회
- Nam Kyun Kim, "Armendariz Rings and Reduced Rings", Journal of Algebra, 223(2), pp.477~488 (2000), Elsevier BV[DOI]
- N. K. Kim, "On right quasi-duo rings which are -regular", Bull. Korean Math. Soc, 37(2), pp.217~227 (2000)
- Nam Kyun Kim, "Extensions of reversible rings", Journal of Pure and Applied Algebra, 185(1-3), pp.207~223 (2003), Elsevier BV[DOI]
- T. K. Kwak, "Quasi-normality of idempotents on nilpotents", Hacettepe J. Math. Stat
- Greg Marks, "ON 2-PRIMAL ORE EXTENSIONS", Communications in Algebra, 29(5), pp.2113~2123 (2001), Informa UK Limited[DOI]
- N. H. McCoy, "Generalized regular rings", Bulletin of the American Mathematical Society, 45(2), pp.175~179 (1939), American Mathematical Society (AMS)[DOI]
- J. von Neumann, "On regular rings", Proceedngs of the National Academy of Sciences, 22, pp.707~713 (1936)
- Agata Smoktunowicz, "Polynomial Rings over Nil Rings Need Not Be Nil", Journal of Algebra, 233(2), pp.427~436 (2000), Elsevier BV[DOI]