Algebras and Representation Theory
, Volume 16, Issue 3, pp 843857
First online:
Classes of Almost Clean Rings
 Evrim AkalanAffiliated withDepartment of Mathematics, Hacettepe University
 , Lia VašAffiliated withDepartment of Mathematics, Physics and Statistics, University of the Sciences Email author
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A ring is clean (almost clean) if each of its elements is the sum of a unit (regular element) and an idempotent. A module is clean (almost clean) if its endomorphism ring is clean (almost clean). We show that every quasicontinuous and nonsingular module is almost clean and that every right CS (i.e. right extending) and right nonsingular ring is almost clean. As a corollary, all right strongly semihereditary rings, including finite AW ^{*}algebras and noetherian Leavitt path algebras in particular, are almost clean. We say that a ring R is special clean (special almost clean) if each element a can be decomposed as the sum of a unit (regular element) u and an idempotent e with aR ∩ eR = 0. The CamilloKhurana Theorem characterizes unitregular rings as special clean rings. We prove an analogous theorem for abelian Rickart rings: an abelian ring is Rickart if and only if it is special almost clean. As a corollary, we show that a right quasicontinuous and right nonsingular ring is left and right Rickart. If a special (almost) clean decomposition is unique, we say that the ring is uniquely special (almost) clean. We show that (1) an abelian ring is unitregular (equiv. special clean) if and only if it is uniquely special clean, and that (2) an abelian and right quasicontinuous ring is Rickart (equiv. special almost clean) if and only if it is uniquely special almost clean. Finally, we adapt some of our results to rings with involution: a *ring is *clean (almost *clean) if each of its elements is the sum of a unit (regular element) and a projection (selfadjoint idempotent). A special (almost) *clean ring is similarly defined by replacing “idempotent” with “projection” in the appropriate definition. We show that an abelian *ring is a Rickart *ring if and only if it is special almost *clean, and that an abelian *ring is *regular if and only if it is special *clean.
Keywords
Clean Almost clean Quasicontinuous Nonsingular Rickart Abelian and CS ringsMathematics Subject Classifications (2010)
16U99 16W99 16W10 16S99 Title
 Classes of Almost Clean Rings
 Journal

Algebras and Representation Theory
Volume 16, Issue 3 , pp 843857
 Cover Date
 201306
 DOI
 10.1007/s1046801293346
 Print ISSN
 1386923X
 Online ISSN
 15729079
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

 Clean
 Almost clean
 Quasicontinuous
 Nonsingular
 Rickart
 Abelian and CS rings
 16U99
 16W99
 16W10
 16S99
 Authors

 Evrim Akalan ^{(1)}
 Lia Vaš ^{(2)}
 Author Affiliations

 1. Department of Mathematics, Hacettepe University, Beytepe Campus, Ankara, 06532, Turkey
 2. Department of Mathematics, Physics and Statistics, University of the Sciences, Philadelphia, PA, 19104, USA