On two questions of Nicholson
We show that a ring $R$ has stable range one if and only if every left unit lifts modulo every left principal ideal. We also show that a left quasi-morphic ring has stable range one if and only if it is left uniquely generated. Thus we answer in the affirmative the two questions raised by W. K. Nicholson.
___
- H. Bass, K-theory and stable algebra, Inst. Hautes Etudes Sci. Publ. Math., 22 (1964), 5-60.
- V. Camillo and W. K. Nicholson, Quasi-morphic rings, J. Algebra Appl., 6 (2007), 789-799.
- V. Camillo, W. K. Nicholson and Z. Wang, Left quasi-morphic rings , J. Algebra Appl., 7(6) (2008), 725-733.
- V. P. Camillo and H. P. Yu, Exchange rings, units and idempotents, Comm. Algebra, 22(12) (1994), 4737-4749.
- M. J. Canfell, Completion of diagram by automorphisms and Bass's first stable range condition, J. Algebra, 176 (1995), 480-503.
- M. J. Canfell Uniqueness of generators of principal ideals in rings of continuous function, Proc. Amer. Math. Soc., 26 (1970), 517-573.
- L. Fuchs, On a substitution property of modules, Monatsh. Math., 75 (1971), 198-204.
- K. R. Goodearl, Cancellation of low rank vector bundles, Pacific. J. Math., 113(2) (1984), 289-302.
- K. R. Goodearl, Von Neumann Regular Rings, Second ed., Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1991.
- R. Hartwig and J. Luh, A note on the group structure of unit regular ringelements, Pacific J. Math., 71 (1977), 449-461.
- M. Henriksen, On a class of regular rings that are elementary divisor rings, Arch. Math. (Basel), 24 (1973), 133-141.
- M. Henriksen, Two classes of rings generated by their units, J. Algebra, 31 (1974), 182-193.
- I. Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc., 66 (1949), 464-491.
- I. Kaplansky, Bass's First Stable Range Condition, Mimeographed Notes, 1971.
- T. Y. Lam, A crash course on stable range, cancellation, substitution, and exchange, J. Algebra Appl., 3 (2004), 301-343.
- P. Menal and J. Moncasi, Lifting units in self-injective rings and an indextheory for Rickart C*-algebras, Pacific. J. Math., 126(2) (1987), 295-329.
- W. K. Nicholson, Lifting idempotents and exchange Rings, Trans. Amer. Math. Soc., 229 (1977), 269-278.
- L. N. Vaserstein, Bass's first stable range condition, J. Pure and Appl. Algebra, 34 (1984), 319-330.
- L. E. T. Wu and J. P. Jans, On quasi projectives, Illinois J. Math., 11 (1967), 439-448.
- X. Yang, On rings whose finitely generated left ideals are left annihilators ofan element, arXiv:1002.3193v2 [math.RA], 28 Apr 2010.
- H. P. Yu, Stable range one for exchange rings, J. Pure Appl. Algebra, 98 (1995), 105-109.
- ISSN: 1306-6048
- Yayın Aralığı: Yılda 2 Sayı
- Başlangıç: 2007
- Yayıncı: Abdullah HARMANCI