On two questions of Nicholson

Year 2022, Volume: 31 Issue: 31, 90 - 99, 17.01.2022

Feroz Sıddıque

https://doi.org/10.24330/ieja.1058418

Abstract

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.

Keywords

Directly-finite ring, stable range one, left quasi-morphic ring

References

• 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.