Abstract
We prove that every exchange ring with primitive factors Artinian is clean. Also, it is shown that for exchange rings with Artinian primitive factors, the following are equivalent: (1) Every element in R is a sum of two units. (2) There exist α, β ∈ U(R) such that α + β = 1. (3) R does not have Z / 2 Z as a homomorphic image. Finally, we prove that exchange ring R is strongly π-regular if the Jacobson radical of any homomorphic image of R is T-nilpotent or locally nilpotent. These are generalizations of the corresponding results of A. Badawi, W. D. Burgess and P. Menal, Fisher and Snider, and J. Stock.
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References
Badawi, A.: On Abelian π-regular rings, Comm. Algebra 25 (1997), 1009-1021.
Burgess, W. D. and Menal, P.: On strongly π-regular rings and homomorphisms into them, Comm. Algebra 16 (1988), 1701-1725.
Camillo, P. and Yu, H. P.: Exchange rings, units and idempotents, Comm. Algebra 22 (1994), 4737-4749.
Chen, H.: Elements in one-sided unit-regular rings, Comm. Algebra 25 (1997), 2517-2529.
Chen, H.: Exchange rings, related comparability and power-substitution, Comm. Algebra 26 (1998), 3383-3401.
Fisher, J. W. and Snider, R. L.: On the von Neumann regularity of rings with regular prime factor rings, Pacific J. Math. 54 (1974), 135-144.
Fisher, J. W. and Snider, R. L.: Rings generated by their units, J. Algebra 42 (1976), 363-368.
Goodearl, K. R.: Von Neumann Regular Rings, Pitman, London, 1979.
Lam, T. Y.: A First Course in Noncommutative Rings, Springer-Verlag, Beijing, 1997.
Menal, P.: On π-regular rings whose primitive factors are Artinian, J. Pure Appl. Algebra 20 (1981), 71-81.
Nicholson, W. K.: Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977), 269-278.
Stock, J.: On rings whose projective modules have exchange property, J. Algebra 103 (1986), 437-453.
Yu, H. P.: On quasi-duo rings, Glasgow Math. J. 37 (1995), 21-31.
Yu, H. P.: Stable range one for exchange rings, J. Pure Appl. Algebra 98 (1995), 105-109.
Yu, H. P.: On the structure of exchange rings, Comm. Algebra 25 (1997), 661-670.
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Chen, H. Exchange Rings with Artinian Primitive Factors. Algebras and Representation Theory 2, 201–207 (1999). https://doi.org/10.1023/A:1009927211591
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DOI: https://doi.org/10.1023/A:1009927211591