Classes of Almost Clean Rings
 Evrim Akalan,
 Lia Vaš
 … show all 2 hide
Purchase on Springer.com
$39.95 / €34.95 / £29.95*
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Abstract
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.
 Abrams, G., Rangaswamy, K.L.: Regularity conditions for arbitrary Leavitt path algebras. Algebr. Represent. Theory 13(3), 319–334 (2010) CrossRef
 Aranda Pino, G., Pardo, E., Siles Molina, M.: Exchange Leavitt path algebras and stable rank. J. Algebra 305(2), 912–936 (2006) CrossRef
 Beidar, K.I., Jain, S.K., Kanwar, P.: Nonsingular CSrings coincide with tight PPrings. J. Algebra 282, 626–637 (2004) CrossRef
 Camillo, V.P., Khurana, D.: A characterization of unit regular rings. Commun. Algebra 29(5), 2293–2295 (2001) CrossRef
 Camillo, V.P., Khurana, D., Lam, T.Y., Nicholson, W.K., Zhou, Y.: Continuous modules are clean. J. Algebra 304, 94–111 (2006) CrossRef
 Chatters, A.W., Hajarnavis, C.R.: Rings in which every complement right ideal is a direct summand. Q. J. Math. Oxford 2(28), 61–80 (1977) CrossRef
 Endo, S.: Note on p.p. rings (a supplement to Hattoris paper). Nagoya Math. J. 17, 167–170 (1960)
 Goel, V.K., Jain, S.K.: πinjective modules and rings whose cyclics are πinjective. Commun. Algebra 6(1), 59–73 (1978) CrossRef
 Goodearl, K.R.: Von Neumann Regular Rings, 2nd edn. Krieger, Malabar (1991)
 Lam, T.Y.: Lectures on Modules and Rings. Graduate Texts in Mathematics, vol. 189. Springer, New York (1999) CrossRef
 Lee, G., Rizvi, S.T., Roman, C.S.: Rickart modules. Commun. Algebra 38(11), 4005–4027 (2010) CrossRef
 Lee, T.K., Yi, Z., Zhou, Y.: An example of Bergman’s and the extension problem for clean rings. Commun. Algebra 36, 1413–1418 (2008) CrossRef
 McGovern, W.Wm.: Clean semiprime frings with bounded inversion. Commun. Algebra 31(7), 3295–3304 (2003) CrossRef
 Mohamed, S.H., Müller, B.J.: Continuous and Discrete Modules. London Math. Soc. Lec. Notes Series, vol. 147. Cambridge University Press (1990) CrossRef
 Nicholson, W.K., Varadarajan, K.: Countable linear transformations are clean. Proc. Am. Math. Soc. 126(1), 61–64 (1998) CrossRef
 Vaš, L.: Dimension and torsion theories for a class of Baer *rings. J. Algebra 289(2), 614–639 (2005) CrossRef
 Vaš, L.: *clean rings; cleanness of some Baer *rings and von Neumann algebras. J. Algebra 324(12), 3388–3400 (2010) CrossRef
 Vaš, L.: Strongly semihereditary rings and rings with dimension. Algebr. Represent. Theory (in print). doi:10.1007/s1046801192791
 Zhu, H., Ding, N.L.: Generalized morphic rings and their applications. Commun. Algebra 35, 2820–2837 (2007) CrossRef
 Title
 Classes of Almost Clean Rings
 Journal

Algebras and Representation Theory
Volume 16, Issue 3 , pp 843857
 Cover Date
 20130601
 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