Special regular clean rings

Document Type : Research Paper

Authors

Department of Mathematics, Gauhati University, Jalukbari, Guwahati, India.

Abstract

In this paper we introduce the concepts of special regular clean elements and regular clean decomposition in a ring R. These concepts lead us to the notion of special regular clean ring. We prove that for a special regular clean element $a=e+r \in R$ and unit $u\in R$ then $au$ is a special regular clean if $u$ is an inner inverse of $e$. We establish that an abelian ring $R$ is a special regular clean ring if and only if the twisted power series ring $R[[x,\sigma]]$ is a special regular clean ring. We also study various characterizations of special clean and special regular clean rings.

Keywords

References

[1] D. D. Anderson and V. P. Camillo, Commutative rings whose elements are a sum of a unit and idempotent, Commun. Algebra, 30 No. 7 (2002) 3327-3336.
[2] P. N. Ánh, G. F. Birkenmeier and L. van Wyk, Peirce decompositions, idempotents and rings, J. Algebra, 564 (2020) 247-275.
[3] N. Ashrafi and E. Nasibi, r-clean rings, Math. Rep., 15 (2011) 125-132.
[4] N. Ashrafi and E. Nasibi, Rings in which elements are the sum of an idempotent and a regular element, Bull. Iran. Math. Soc., 39 No. 3 (2013) 579-588.
[5] V. P. Camillo and H. P. Yu. Exchange rings, units and idempotents, Communications in Algebra, 22 No. 12 (1994) 4737-4749.
[6] J. Chen, W. K. Nicholson and Y. Zhou, Group rings in which every element is uniquely the sum of a unit and an idempotent, J. Algebra, 306 No. 2 (2006) 453-460.
[7] W. Chen, Notes on clean rings and clean elements, Southeast Asian Bulletin of Mathematics, 2998 No. 32 (2008) 00-6.
[8] K. R. Goodearl and R. B. Warfield, An Introduction to Noncommutative Noetherian Rings, Cambridge University Press, 2004.
[9] K. R. Goodea,rl. Von Neumann Regular Rings, Pitman, 1979.
[10] Juncheol Han and W. Keith Nicholson. Extensions of clean rings, Commun. Algebra, 29 No. 6 (2001) 2589-2595.
[11] D. Handelman, Perspectivity and cancellation in regular rings, J. Algebra, 48 No. 1 (1977) 1-16.
[12] D. Khurana, T. Y. Lam, P. P. Nielsen, and J. Šter, Special clean elements in rings, J. Algebra Its Appl., 19 No. 11 (2020) 2050208, 2020.
[13] T. Y. Lam. A First Course in Noncommutative Rings, Volume 131, Springer, 1991.
[14] T. Y. Lam and P. P. Nielsen, Inner inverses and inner annihilators in rings, J. Algebra, 397 (2014) 91-110.
[15] W. W. McGovern, A characterization of commutative clean rings, Int. J. Math. Game Theory Algebra, 15 No. 4 (2006) 403-413.
[16] W. K. Nicholson, Lifting idempotents and exchange rings, T. AM. MATH. SOC., 229 (1977) 269-278.
[17] W. K. Nicholson and Y. Zhou, Rings in which elements are uniquely the sum of an idempotent and a unit, Glasg. Math. J., 46 No. 2 (2004) 227-236.
Algebraic Structures and Their Applications
Volume 12, Issue 1
February 2025
Pages 23-32

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