Abstract
Let R be a ring with identity and J(R) denote the Jacobson radical of R. A ring R is called J-reversible if for any a, \(b \in R\), \(ab = 0\) implies \(ba \in J(R)\). In this paper, we give some properties of J-reversible rings. We prove that some results of reversible rings can be extended to J-reversible rings for this general setting. We show that J-quasipolar rings, local rings, semicommutative rings, central reversible rings and weakly reversible rings are J-reversible. As an application it is shown that every J-clean ring is directly finite.
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Calci, M.B., Halicioglu, S., Harmanci, A.: A class of \(J\)-quasipolar rings. J. Algebra Relat. Top. 3(2), 1–15 (2015)
Camillo, V.P., Yu, H.P.: Exchange rings, units and idempotents. Commun. Algebra 22(12), 4737–4749 (1994)
Chen, H.: On strongly J-clean rings. Commun. Algebra 38, 3790–3804 (2010)
Chen, H.: Rings Related Stable Range Conditions. Series in algebra, vol. 11. World Scientific, Hackensack (2011)
Chen, H., Gurgun, O., Halicioglu, S., Harmanci, A.: Rings in which nilpotents belong to Jacobson radical. An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) LXII(2), 595–606 (2016)
Cohn, P.M.: Reversible rings. Bull. Lond. Math. Soc. 31(6), 641–648 (1999)
Cui, J., Chen, J.: A class of quasipolar rings. Commun. Algebra 40, 4471–4482 (2012)
Koliha, J.J., Patricio, P.: Elements of rings with equal spectral idempotents. J. Aust. Math. Soc. 72(1), 137–152 (2002)
Kose, H., Ungor, B., Halicioglu, S., Harmanci, A.: A generalization of reversible rings. Iran. J. Sci. Technol. Trans. A Sci. 38(1), 43–48 (2014)
Krylov, P.A.: Isomorphism of generalized matrix rings. Algebra Log. 47(4), 258–262 (2008)
Lam, T.Y., Dugas, A.S.: Quasi-duo rings and stable range descent. J. Pure Appl. Algebra 195, 243–259 (2005)
Lambek, J.: On the representation of modules by sheaves of factor modules. Can. Math. Bull. 14, 359–368 (1971)
Liang, Z., Gang, Y.: On weakly reversible rings. Acta Math. Univ. Comen. 76(2), 189–192 (2007)
Marianne, M.: Rings of quotients of generalized matrix rings. Commun. Algebra 15, 1991–2015 (1987)
Morita, K.: Duality for modules and its applications to the theory of rings with minimum condition. Sci. Rep. Tokyo Kyoiku Diagaku Sect. A 6, 83–142 (1958)
Nicholson, W.K.: Lifting idempotents and exchange rings. Trans. Am. Math. Soc. 229, 269–278 (1977)
Nicholson, W.K.: Strongly clean rings and Fitting’s lemma. Commun. Algebra 27(8), 3583–3592 (1999)
Tang, G., Li, C., Zhou, Y.: Study of Morita contexts. Commun. Algebra 42(4), 1668–1681 (2014)
Warfield, R.B.: Exchange rings and decompositions of modules. Math. Ann. 199, 31–36 (1972)
Yu, H.P.: Stable range one for exchange rings. J. Pure Appl. Algebra 98, 105–109 (1995)
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Calci, M.B., Chen, H., Halicioglu, S. et al. Reversibility of Rings with Respect to the Jacobson Radical. Mediterr. J. Math. 14, 137 (2017). https://doi.org/10.1007/s00009-017-0938-2
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DOI: https://doi.org/10.1007/s00009-017-0938-2