A note on uniquely (nil) clean ring

Subject Areas : History and biography

Sh. Sahebi ¹ , M. Jahandar ^{2 *}

1 - Department of Mathematics, Islamic Azad University, Central Tehran Branch, PO. Code 14168-94351, Iran
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.

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