Quasi regular modules and trivial extension

Year 2021, Volume: 50 Issue: 1, 120 - 134, 04.02.2021

Chillumuntala Jayaram Ünsal Tekir , Suat Koç

https://doi.org/10.15672/hujms.613404

Cited By: 7

Abstract

Recall that a ring $R\ $is said to be a quasi regular ring if its total quotient ring $q(R)\ $is \textit{von Neumann regular}. It is well known that a ring $R\ $is quasi regular if and only if it is a reduced ring satisfying the property: for each $a\in R,$ $ann_{R}(ann_{R}(a))=ann_{R}(b)$ for some $b\in R$. Here, in this study, we extend the notion of quasi regular rings and rings which satisfy the aforementioned property to modules. We give many characterizations and properties of these two classes of modules. Moreover, we investigate the (weak) quasi regular property of trivial extension.

Keywords

von Neumann regular rings, quasi regular rings, von Neumann regular module, quasi regular module, trivial extension

References

• [1] D.D. Anderson and M. Winders, Idealization of a module, J. Commut. Algebra, 1 (1), 3–56, 2009.
• [2] D.F. Anderson, R. Levy and J. Shapiro, Zero-divisor graphs, von Neumann regular rings, and Boolean algebras, J. Pure Appl. Algebra, 180 (3), 221–241, 2003.
• [3] Z.A. El-Bast and P.F. Smith, Multiplication modules, Comm. Algebra, 16 (4), 755– 779, 1988.
• [4] M. Evans, On commutative P.P. rings, Pac. J. Math. 41 (3), 687–697, 1972
• [5] M. Henriksen and M. Jerison, The space of minimal prime ideals of a commutative ring, Trans. Amer. Math. Soc. 115, 110–130, 1965.
• [6] J.A. Huckaba, Commutative rings with zero divisors, Marcel Dekker, New York, 1988.
• [7] C. Jayaram, Baer ideals in commutative semiprime rings, Indian J. Pure Appl. Math. 15 (8) 855–864, 1984.
• [8] C. Jayaram and Ü. Tekir, von Neumann regular modules, Comm. Algebra, 46 (5), 2205–2217, 2018.
• [9] T.K. Lee and Y. Zhou, Reduced modules, Rings, modules, algebras and abelian groups, in:Lect. Notes Pure Appl. Math. New York, NY: Marcel Dekker, 236, 365–377, 2004.
• [10] R. Levy and J. Shapiro, The zero-divisor graph of von Neumann regular rings, Comm. Algebra, 30 (2), 745–750, 2002.
• [11] C.P. Lu, Prime submodules of modules, Comment. Math. Univ. St. Pauli, 33 (1), 61–69, 1984.
• [12] R.L. McCasland and M.E. Moore, Prime submodules, Comm. Algebra, 20 (6), 1803– 1817, 1992.
• [13] M. Nagata, Local rings, Interscience Publishers, New York, 1960.
• [14] J. Von Neumann, On regular rings, Proc. Natl. Acad. Sci. 22 (12), 707–713, 1936.