Left Nil Zero Semicommutative rings
Abstract
This paper introduces a class of rings called left nil zero semicommutative rings ( LNZS rings ), wherein a ring R is said to be LNZS if the left annihilator of every nilpotent element of R is an ideal of R. It is observed that reduced rings are LNZS but not the other way around. So, this paper provides some conditions for an LNZS ring to be reduced and among other results, it is proved that R is reduced if and only if the ring of upper triangular matrices over R is LNZS. Furthermore, it is shown that the polynomial ring over an LNZS may not be LNZS and so is the case of the skew polynomial over an LNZS ring. Therefore, this paper investigates the LNZS property over the polynomial extension and skew polynomial extension of an LNZS ring with some additional conditions.
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References
Chen, W., On nil-semicommutative rings, Thai. J. Math., 9, 39-47, (2011).
Kim, N. K., Lee, Y., Extensions of reversible rings, J. Pure Appl. Algebra, 185, no. 1-3, 207-223, (2003). DOI: https://doi.org/10.1016/S0022-4049(03)00109-9
Liang, L., Wang, L., Liu, Z., On a generalization of semicommutative rings, Taiwan. J. Math, 11, no. 5, 1359-1368, (2007). DOI: https://doi.org/10.11650/twjm/1500404869
Mohammadi, R., Moussavi, A., Zahiri, M., On nil-semicommutative rings, Int. Electron. J. Algebra, 11, no. 11, 20-37, (2012).
Ozen, T., Agayev, N., Harmanci, A., ¨ On a class of semicommutative rings, Kyungpook Math. J., 51, no. 3, 283-291, (2011). DOI: https://doi.org/10.5666/KMJ.2011.51.3.283
Rege, M. B., Chhawchharia, S., Armendariz rings, Proc. Japan Acad., 73, 14-17, (1997). DOI: https://doi.org/10.3792/pjaa.73.14
Wei, J., Certain rings whose simple singular modules are nil-injective, Turk. J. Math, 32, no. 4, 393-408, (2008).
Wei, J. C., Chen, J. H., Nil-injective rings, Int. Electron. J. Algebra, 2, no. 2, 1-21, (2007).
Wei, J., Li, L., Quasi-normal rings, Comm. Algebra, 38, no. 5, 1855-1868, (2010). DOI: https://doi.org/10.1080/00927871003703943
Wei-xing, C., Shu-ying, C., On weakly semicommutative rings, Comm. Math. Res., 27, no. 2, 179-192, (2011).
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