RINGS CLOSE TO SEMIREGULAR

Title & Authors
RINGS CLOSE TO SEMIREGULAR
Aydogdu, Pinar; Lee, Yang; Ozcan, A. Cigdem;

Abstract
A ring $\small{R}$ is called semiregular if $\small{R/J}$ is regular and idem-potents lift modulo $\small{J}$, where $\small{J}$ denotes the Jacobson radical of $\small{R}$. We give some characterizations of rings $\small{R}$ such that idempotents lift modulo $\small{J}$, and $\small{R/J}$ satisfies one of the following conditions: (one-sided) unit-regular, strongly regular, (unit, strongly, weakly) $\small{{\pi}}$-regular.
Keywords
idempotent lifting;semi unit-regular ring;semi (strongly) $\small{{\pi}}$-regular ring;
Language
English
Cited by
References
1.
P. Ara, Strongly $\pi$-regular rings have stable range one, Proc. Amer. Math. Soc. 124 (1996), no. 11, 3293-3298.

2.
R. F. Arens and I. Kaplansky, Topological representations of algebras, Trans. Amer. Math. Soc. 63 (1948), 457-481.

3.
G. Azumaya, Strongly $\pi$-regular rings, J. Fac. Sci. Hokkaido Univ. Ser. I 13 (1954), 34-39.

4.
V. P. Camillo and H. P. Yu, Stable range one for rings with many idempotents, Trans. Amer. Math. Soc. 347 (1995), no. 8, 3141-3147.

5.
V. P. Camillo and H. P. Yu, Exchange rings, units and idempotents, Comm. Algebra 22 (1994), no. 12, 4737-4749.

6.
H. Chen, A note on the $\pi$-regularity of rings, Chinese Quart. J. Math. 13 (1998), no. 2, 67-71.

7.
H. Chen, Exchange rings satisfying unit 1-stable range, Kyushu J. Math. 54 (2000), 1-6.

8.
H. Chen, On strongly stable rings, Comm. Algebra 31 (2003), no. 6, 2771-2789.

9.
H. Chen and M. Chen, On semiregular rings, New Zealand J. Math. 32 (2003), 11-20.

10.
A. Y. M. Chin, A note on strongly $\pi$-regular rings, Acta Math. Hungar. 102 (2004), no. 4, 337-342.

11.
P. Crawley and B. Jonsson, Refinements for infinite direct decompositions of algebraic systems, Pacific J. Math. 14 (1964), 797-855.

12.
K. R. Goodearl, Von-Neumann Regular Rings, Pitman Publishing Limited, London, 1979.

13.
V. Gupta, Weakly $\pi$-regular rings and group rings, Math. J. Okayama Univ. 19 (1977), no. 2, 123-127.

14.
Y. Hirano, Some studies on strongly $\pi$-regular rings, Math. J. Okayama Univ. 20 (1978), no. 2, 141-149.

15.
C. H. Hong, N. K. Kim, T. K. Kwak, and Y. Lee, On weak $\pi$-regularity of rings whose prime ideals are maximal, J. Pure Appl. Algebra 146 (2000), no. 1, 35-44.

16.
Q. Huang and J. Chen, $\pi$-morphic rings, Kyungpook Math. J. 47 (2007), no. 3, 363-372.

17.
S. U. Hwang, Y. C. Jeon, and Y. Lee, Structure and topological conditions of NI-rings, J. Algebra 302 (2006), no. 1, 186-199.

18.
Q. Li and W. T. Tong, Weak cancellation of modules and weakly stable range conditions for exchange rings, Acta Math. Sinica 45 (2002), no. 6, 1121-1126.

19.
W. K. Nicholson, Strongly clean rings and fitting's lemma, Comm. Algebra, 27 (1999), no. 8, 3583-3592.

20.
W. K. Nicholson and M. F. Yousif, Quasi-Frobenius Rings, Cambridge Tracts in Math. 158., Cambridge University Press, Cambridge, UK, 2003.

21.
W. K. Nicholson and Y. Zhou, Strong lifting, J. Algebra 285 (2005), no. 2, 795-818.

22.
V. S. Ramamurthi, Weakly regular rings, Canad. Math. Bull. 16 (1973), 317-321.

23.
A. A. Tuganbaev, Semiregular, weakly regular and $\pi$-regular rings, J. Math. Sci. (New York) 109 (2002), no. 3, 1509-1588.

24.
L. N. Vaserstein, Bass's first stable range condition, J. Pure Appl. Algebra 34 (1984), no. 2-3, 319-330.

25.
R. B. Warfield, Exchange rings and decomposition of modules, Math. Ann. 199 (1972), 31-36.

26.
T. S. Wu, Weak cancellation of modules and the weak stable range one condition, Nanjing Daxue Xuebao Shuxue Bannian Kan 11 (1994), no. 2, 109-116.

27.
T. S. Wu, Inner weak cancellation of modules, Nanjing Daxue Xuebao Shuxue Bannian Kan 13 (1996), no. 1, 54-57.

28.
G. Xiao and W. Tong, Generalizations of semiregular rings, Comm. Algebra 33 (2005), no. 10, 3447-3465.

29.
H. P. Yu, On quasi-duo rings, Glasg. Math. J. 37 (1995), no. 1, 21-31.

30.
H. P. Yu, On the structure of exchange rings, Comm. Algebra 25 (1997), no. 2, 661-670.