A note on uniquely (nil) clean ring

Subject Areas : History and biography

Sh. Sahebi ¹ (Department of Mathematics, Islamic Azad University, Central Tehran Branch, PO. Code 14168-94351, Iran)
M. Jahandar ² (Department of Mathematics, Islamic Azad University, Central Tehran Branch, PO. Code 14168-94351, Iran)

Received: 2012-01-01 Accepted : 2012-04-01 Published : 2012-06-01

Keywords: Full element, uniquely clean ring, nil clean ring,

Abstract :

A ring R is uniquely (nil) clean in case for any $a \in R$ there exists a uniquelyidempotent $e\in R$ such that $a-e$ is invertible (nilpotent). Let $C =(A VW B)$be the MoritaContext ring. We determine conditions under which the rings $A,B$ are uniquely (nil) clean.Moreover we show that the center of a uniquely (nil) clean ring is uniquely (nil) clean.

References:

[1] M. Y. Ahn, (2003). Weakly clean rings and almost clean rings. Ph.D. Thesis, University of Lowa.

[2] D. D. Anderson, V. P. Camillo, Commutative rings whose elements are a sum of unit and idempotent. Comm. Algebra 30 (2002), pp. 3327-3336.

[3] B. Li, L. Feng, F-clean rings and rings having many full elements. J. Korean Math. Soc. 2 (2010), pp. 247-261.

[4] J. Che, W. K. Nicholson, Y. Zhou, Group rings in which every element is uniquely the sum of a unit and idempotent. J. Algebra. 306 (2006), pp. 453-460.

[5] H. Chen, Morita contexts with many units. Comm. Algebra. 30 (3) (2002), pp. 1499-1512.

[6] A. J. Diesl, Classes of strongly clean rings. Ph.D. Thesis, University of California, Berkeley, (2006).

[7] A. Haghany, Hopcity and co-hopcity for Morita Contexts. Comm. Algebra. 27(1)(1999), pp. 477-492.

[8] W. K. Nicholson, Y. Zhou, Rings in which elements are uniquely the some of an idempotent and unit. Clasy. Math. J. 46(2004), pp. 227-236.