Burgess, W. D. and Raphael, Robert (2013) On Embedding Rings in Clean Rings. Communications in Algebra, 41 (2). pp. 552-564. ISSN 0092-7872
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Official URL: http://dx.doi.org/10.1080/00927872.2011.607873
Abstract
A clean ring is one in which every element is a sum of an idempotent and a unit. It is shown that every ring can be embedded in a clean ring as an essential ring extension. It is seen that the centre of a clean ring need not be a clean ring. There is no “clean hull” of a ring. A family of examples is given where there is a ring R, not a clean ring, embedded in a commutative clean ring S so that there is no clean ring T, R T S, minimal with that property. It is also shown that a commutative pm ring (each prime ideal is contained in a unique maximal ideal) cannot be extended to a clean ring by the adjunction of finitely many central idempotents.
| Divisions: | Concordia University > Faculty of Arts and Science > Mathematics and Statistics |
|---|---|
| Item Type: | Article |
| Refereed: | Yes |
| Authors: | Burgess, W. D. and Raphael, Robert |
| Journal or Publication: | Communications in Algebra |
| Date: | 2013 |
| Digital Object Identifier (DOI): | 10.1080/00927872.2011.607873 |
| Keywords: | Clean ring, Local ring, Rings of continuous functions |
| ID Code: | 976886 |
| Deposited By: | Danielle Dennie |
| Deposited On: | 14 Feb 2013 19:00 |
| Last Modified: | 18 Jan 2018 17:43 |
References:
1. Anderson , D. D. , Camillo , V. P. ( 2002 ). Commutative rings whose elements are a sum of a unit and idempotent . Comm. Algebra 30 : 3327 – 3336 .2. Bergman , G. ( 1971 ). Hereditary commutative rings and centres of hereditary rings . Proc. London Math. Soc. 23 : 214 – 236 .
3. Burgess , W. D. , Raphael , R. ( 2009 ). On commutative clean rings and pm rings . Contemp. Math. 480 : 35 – 55 .
4. Burgess , W. D. , Stephenson , W. ( 1979 ). Rings all of whose Pierce stalks are local . Canad. Math. Bull. 22 : 159 – 164 .
5. Camillo , V. P. , Khurana , D. , Lam , T. Y. , Nicholson , W. K. , Zhou , Y. ( 2006 ). Continuous modules are clean . J. Algebra 304 : 94 – 111 .
6. Camillo , V. P. , Yu , H.-P. ( 1994 ). Exchange rings, units and idempotents . Comm. Algebra 22 : 4737 – 4749 .
7. Contessa , M. ( 1982 ). On PM-rings . Comm. Algebra 10 : 93 – 108 .
8. De Marco , G. , Orsatti , A. (1971). Commutative rings in which every prime ideal is contained in a unique maximal ideal. Proc. Amer. Math. Soc. 30:459–466.
9. Feigelstock , S. ( 1983 ). Additive Groups of Rings. Vol. 1. Pitman Research Notes in Mathematics, 83, Longman .
10. Feigelstock , S. ( 1988 ). Additive Groups of Rings. Vol. II. Pitman Research Notes in Mathematics, 169, Longman .
11. Fine , N. J. , Gillman , L. , Lambek , J. ( 2005 ). Rings of Quotients of Rings of Functions. Lecture Notes in Real Algebraic and Analytic Geometry, RAAG .
12. Gillman , L. , Jerison , M. ( 1976 ). Rings of Continuous Functions . Grad. Texts in Math . Vol. 43 . Berlin : Springer-Verlag .
13. Goodearl , K. R. , Warfield , Jr. , R. B. ( 1976 ). Algebras over zero-dimensional rings . Math. Ann. 223 : 157 – 168 .
14. Han , J. , Nicholson , W. K. ( 2001 ). Extensions of clean rings . Comm. Algebra 29 : 2589 – 2595 .
15. Huisgen-Zimmermann , B. ( 2000 ). Purity, Algebraic Compactness, Direct Sum Decompositions, And Representation Type. In Infinite Length Modules, Trends in Mathematics, Birkhäuser .
16. Johnstone , P. T. ( 1992 ). Stone Spaces . Cambridge Studies in Advanced Mathematics . Vol. 3 . Cambridge : Cambridge University Press .
17. Jensen , C. U. , Zimmermann-Huisgen , B. ( 1989 ). Algebraic compactness of ultrapowers and representation type . Pacific J. Math. 139 : 251 – 265 .
18. Lam , T. Y. ( 1998 ). Lectures on Modules and Rings. Graduate Texts in Mathematics Vol. 189. Springer-Verlag .
19. Lu , D. C. , Yu , W. H. ( 2006 ). On prime spectrums of commutative rings . Comm. Algebra 34 : 2667 – 2672 .
20. McGovern , W. Wm. ( 2003 ). Clean semiprime f-rings with bounded inversion . Comm. Algebra 31 : 3295 – 3304 . [Taylor & Francis Online],
21. Mohamed , S. H. , Müller , B. J. ( 1990 ). Continuous and Discrete Modules. London Math. Soc. Lecture Note Series. Vol. 147. Cambridge .
22. Nicholson , W. K. , Varadarajan , K. , Zhou , Y. ( 2004 ). Clean endomorphism rings . Arch. Math. 83 : 340 – 342 .
23. Nicholson , W. K. ( 1977 ). Lifting idempotents and exchange rings . Trans. Amer. Math. Soc. 229 : 269 – 278 .
24. Nicholson , W. K. , Zhou , Y. ( 2004 ). Rings in which elements are uniquely the sum of an idempotent and a unit . Glasg. Math. J. 46 : 227 – 236 .
25. Pierce , R. S. ( 1967 ). Modules over Commutative Regular Rings. Mem. Amer. Math. Soc. Vol. 70. Providence, RI .
26. Wang , Z. , Chen , J. L. ( 2009 ). 2-clean rings . Canad. Math. Bull. 52 : 145 – 153 .
27. Woods , R. G. ( 2009 ). Private Communication .
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