Browse > Article
http://dx.doi.org/10.4134/JKMS.2012.49.3.605

RINGS CLOSE TO SEMIREGULAR  

Aydogdu, Pinar (Department of Mathematics Hacettepe University)
Lee, Yang (Department of Mathematics Education Pusan National University)
Ozcan, A. Cigdem (Department of Mathematics Hacettepe University)
Publication Information
Journal of the Korean Mathematical Society / v.49, no.3, 2012 , pp. 605-622 More about this Journal
Abstract
A ring $R$ is called semiregular if $R/J$ is regular and idem-potents lift modulo $J$, where $J$ denotes the Jacobson radical of $R$. We give some characterizations of rings $R$ such that idempotents lift modulo $J$, and $R/J$ satisfies one of the following conditions: (one-sided) unit-regular, strongly regular, (unit, strongly, weakly) ${\pi}$-regular.
Keywords
idempotent lifting; semi unit-regular ring; semi (strongly) ${\pi}$-regular ring;
Citations & Related Records
 (Related Records In Web of Science)
연도 인용수 순위
  • Reference
1 W. K. Nicholson, Strongly clean rings and fitting's lemma, Comm. Algebra, 27 (1999), no. 8, 3583-3592.   DOI   ScienceOn
2 W. K. Nicholson and M. F. Yousif, Quasi-Frobenius Rings, Cambridge Tracts in Math. 158., Cambridge University Press, Cambridge, UK, 2003.
3 W. K. Nicholson and Y. Zhou, Strong lifting, J. Algebra 285 (2005), no. 2, 795-818.   DOI   ScienceOn
4 V. S. Ramamurthi, Weakly regular rings, Canad. Math. Bull. 16 (1973), 317-321.   DOI
5 A. A. Tuganbaev, Semiregular, weakly regular and $\pi$-regular rings, J. Math. Sci. (New York) 109 (2002), no. 3, 1509-1588.   DOI
6 L. N. Vaserstein, Bass's first stable range condition, J. Pure Appl. Algebra 34 (1984), no. 2-3, 319-330.   DOI   ScienceOn
7 R. B. Warfield, Exchange rings and decomposition of modules, Math. Ann. 199 (1972), 31-36.   DOI
8 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.
9 T. S. Wu, Inner weak cancellation of modules, Nanjing Daxue Xuebao Shuxue Bannian Kan 13 (1996), no. 1, 54-57.
10 G. Xiao and W. Tong, Generalizations of semiregular rings, Comm. Algebra 33 (2005), no. 10, 3447-3465.   DOI   ScienceOn
11 H. P. Yu, On quasi-duo rings, Glasg. Math. J. 37 (1995), no. 1, 21-31.   DOI
12 H. P. Yu, On the structure of exchange rings, Comm. Algebra 25 (1997), no. 2, 661-670.   DOI   ScienceOn
13 H. Chen, On strongly stable rings, Comm. Algebra 31 (2003), no. 6, 2771-2789.   DOI   ScienceOn
14 V. P. Camillo and H. P. Yu, Exchange rings, units and idempotents, Comm. Algebra 22 (1994), no. 12, 4737-4749.   DOI   ScienceOn
15 H. Chen, A note on the $\pi$-regularity of rings, Chinese Quart. J. Math. 13 (1998), no. 2, 67-71.
16 H. Chen, Exchange rings satisfying unit 1-stable range, Kyushu J. Math. 54 (2000), 1-6.   DOI   ScienceOn
17 H. Chen and M. Chen, On semiregular rings, New Zealand J. Math. 32 (2003), 11-20.
18 A. Y. M. Chin, A note on strongly $\pi$-regular rings, Acta Math. Hungar. 102 (2004), no. 4, 337-342.   DOI
19 P. Crawley and B. Jonsson, Refinements for infinite direct decompositions of algebraic systems, Pacific J. Math. 14 (1964), 797-855.   DOI
20 K. R. Goodearl, Von-Neumann Regular Rings, Pitman Publishing Limited, London, 1979.
21 V. Gupta, Weakly $\pi$-regular rings and group rings, Math. J. Okayama Univ. 19 (1977), no. 2, 123-127.
22 Y. Hirano, Some studies on strongly $\pi$-regular rings, Math. J. Okayama Univ. 20 (1978), no. 2, 141-149.
23 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.   DOI   ScienceOn
24 Q. Huang and J. Chen, $\pi$-morphic rings, Kyungpook Math. J. 47 (2007), no. 3, 363-372.
25 S. U. Hwang, Y. C. Jeon, and Y. Lee, Structure and topological conditions of NI-rings, J. Algebra 302 (2006), no. 1, 186-199.   DOI   ScienceOn
26 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.   DOI   ScienceOn
27 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.
28 P. Ara, Strongly $\pi$-regular rings have stable range one, Proc. Amer. Math. Soc. 124 (1996), no. 11, 3293-3298.   DOI   ScienceOn
29 R. F. Arens and I. Kaplansky, Topological representations of algebras, Trans. Amer. Math. Soc. 63 (1948), 457-481.   DOI   ScienceOn
30 G. Azumaya, Strongly $\pi$-regular rings, J. Fac. Sci. Hokkaido Univ. Ser. I 13 (1954), 34-39.