Generalized semicommutative rings
DOI:
https://doi.org/10.21638/11701/spbu01.2020.110Abstract
We call a ring R generalized semicommutative if for any a, b ∈ R, ab = 0 implies there exists positive integers m, n such that a mRbn = 0. We observe that the class of generalized semicommutative rings strictly lies between the class of central semicommutative rings and weakly semicommutative-I rings. Relationships are provided between generalized semicommutative rings and some known classes of rings. From an arbitrary generalized semicommutative ring, we produce many families of generalized semicommutative rings. Finally we provide some conditions for a generalized semicommutative ring to be reduced.
Keywords:
semicommutative ring, generalized semicommutative ring
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Zhou H. Left SF-rings and regular rings // Comm. Algebra. 2007. Vol. 35. P. 3842–3850.
Liang L., Wang L., Liu Z. On a generalization of semicommutative rings // Taiwanese J. Math. 2007. Vol. 11, no. 5. P. 1359–1368.
Wang L., Wei J. Weakly semicommutative rings and strongly regular rings // Kyungpook Math. J. 2014. Vol. 54. P. 65–72.
Ozen T., Agayev N., Harmanci A. On a class of semicommutative rings // Kyungpook Math. J. 2011. Vol. 51. P. 283–291.
Wang L., Wei J. Central semicommutative rings // Indian J. Pure Appl. Math. 2014. Vol. 45, no. 1. P. 13–25.
Du C., Wang L., Wei J. On a generalization of semicommutative rings // Journal of Mathematical Research and Applications. 2014. Vol. 34, no. 3. P. 253–264.
Liang Z., Gang Y. On weakly reversible rings // Acta Math. Univ. Comenian. (N. S.). 2007. Vol. 76, no. 2. P. 189–182.
Wei J. C., Chen J. H. Nil-injective rings // Intern. Electron. J. Algebra. 2007. Vol. 2. P. 1–21.
Ramamurthy V. S. On the injectivity and flatness of certain cyclic modules // Proc. Amer. Math. Soc. 1975. Vol. 48. P. 21–25.
Rege M. B. On von Neumann regular rings and SF-rings // Math. Japonica. 1986. Vol. 31, no. 6. P. 927–936.
References
Zhou H., “Left SF-rings and regular rings”, Comm. Algebra 35, 3842–3850 (2007).
Liang L., Wang L., Liu Z., “On a generalization of semicommutative rings”, Taiwanese J. Math. 11(5), 1359–1368 (2007).
Wang L., Wei J., “Weakly semicommutative rings and strongly regular rings”, Kyungpook Math. J. 54, 65–72 (2014).
Ozen T., Agayev N., Harmanci A., “On a class of semicommutative rings”, Kyungpook Math. J. 51, 283–291 (2011).
Wang L., Wei J., “Central semicommutative rings”, Indian J. Pure Appl. Math. 45(1), 13–25 (2014).
Du C., Wang L., Wei J., “On a generalization of semicommutative rings”, Journal of Mathematical Research and Applications 34 (3), 253–264 (2014).
Liang Z., Gang Y., “On weakly reversible rings”, Acta Math. Univ. Comenian. (N. S.) 76 (2), 189–182 (2007).
Wei J. C., Chen J. H., “Nil-injective rings”, Intern. Electron. J. Algebra 2, 1–21 (2007).
Ramamurthy V. S., “On the injectivity and flatness of certain cyclic modules”, Proc. Amer. Math. Soc. 48, 21–25 (1975).
Rege M. B., “On von Neumann regular rings and SF-rings”, Math. Japonica 31 (6), 927–936 (1986).
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Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
