Browse > Article
http://dx.doi.org/10.4134/JKMS.j190382

ON LIFTING OF STABLE RANGE ONE ELEMENTS  

Altun-Ozarslan, Meltem (Department of Mathematics Hacettepe University)
Ozcan, Ayse Cigdem (Department of Mathematics Hacettepe University)
Publication Information
Journal of the Korean Mathematical Society / v.57, no.3, 2020 , pp. 793-807 More about this Journal
Abstract
Stable range of rings is a unifying concept for problems related to the substitution and cancellation of modules. The newly appeared element-wise setting for the simplest case of stable range one is tempting to study the lifting property modulo ideals. We study the lifting of elements having (idempotent) stable range one from a quotient of a ring R modulo a two-sided ideal I by providing several examples and investigating the relations with other lifting properties, including lifting idempotents, lifting units, and lifting of von Neumann regular elements. In the case where the ring R is a left or a right duo ring, we show that stable range one elements lift modulo every two-sided ideal if and only if R is a ring with stable range one. Under a mild assumption, we further prove that the lifting of elements having idempotent stable range one implies the lifting of von Neumann regular elements.
Keywords
Stable range one; idempotent stable range one; unit-regular; lifting of units;
Citations & Related Records
연도 인용수 순위
  • Reference
1 F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, second edition, Graduate Texts in Mathematics, 13, Springer-Verlag, New York, 1992. https://doi.org/10.1007/978-1-4612-4418-9
2 G. Baccella, Semi-Artinian V-rings and semi-Artinian von Neumann regular rings, J. Algebra 173 (1995), no. 3, 587-612. https://doi.org/10.1006/jabr.1995.1104   DOI
3 G. Baccella, Exchange property and the natural preorder between simple modules over semi-Artinian rings, J. Algebra 253 (2002), no. 1, 133-166. https://doi.org/10.1016/S0021-8693(02)00044-3   DOI
4 H. Bass, K-theory and stable algebra, Inst. Hautes Etudes Sci. Publ. Math. No. 22 (1964), 5-60.
5 H. Chen, Rings with many idempotents, Int. J. Math. Math. Sci. 22 (1999), no. 3, 547-558. https://doi.org/10.1155/S0161171299225471   DOI
6 D. Estes and J. Ohm, Stable range in commutative rings, J. Algebra 7 (1967), 343-362. https://doi.org/10.1016/0021-8693(67)90075-0   DOI
7 M. A. Fortes Escalona, I. de las Penas Cabrera, and E. Sanchez Campos, Lifting idempotents in associative pairs, J. Algebra 222 (1999), no. 2, 511-523. https://doi.org/10.1006/jabr.1999.8025   DOI
8 L. Fuchs, On a substitution property of modules, Monatsh. Math. 75 (1971), 198-204. https://doi.org/10.1007/BF01299099   DOI
9 S. Garg, H. K. Grover, and D. Khurana, Perspective rings, J. Algebra 415 (2014), 1-12. https://doi.org/10.1016/j.jalgebra.2013.09.055   DOI
10 K. R. Goodearl, von Neumann Regular Rings, second edition, Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1991.
11 D. Khurana and T. Y. Lam, Clean matrices and unit-regular matrices, J. Algebra 280 (2004), no. 2, 683-698. https://doi.org/10.1016/j.jalgebra.2004.04.019   DOI
12 M. Henriksen, On a class of regular rings that are elementary divisor rings, Arch. Math. (Basel) 24 (1973), 133-141. https://doi.org/10.1007/BF01228189   DOI
13 V. A. Hiremath and S. Hegde, Using ideals to provide a unified approach to uniquely clean rings, J. Aust. Math. Soc. 96 (2014), no. 2, 258-274. https://doi.org/10.1017/S1446788713000591   DOI
14 I. Kaplansky, Bass's first stable range condition, mimeographed notes, 1971.
15 D. Khurana and T. Y. Lam, Rings with internal cancellation, J. Algebra 284 (2005), no. 1, 203-235. https://doi.org/10.1016/j.jalgebra.2004.07.032   DOI
16 P. Menal and J. Moncasi, Lifting units in self-injective rings and an index theory for Rickart $C^\ast$-algebras, Pacific J. Math. 126 (1987), no. 2, 295-329. http://projecteuclid.org/euclid.pjm/1102699806   DOI
17 D. Khurana, T. Y. Lam, and P. P. Nielsen, An ensemble of idempotent lifting hypotheses, J. Pure Appl. Algebra 222 (2018), no. 6, 1489-1511. https://doi.org/10.1016/j.jpaa.2017.07.008   DOI
18 T. Y. Lam, A crash course on stable range, cancellation, substitution, and exchange, J. Algebra Appl. 3 (2004), 301-343. https://doi.org/10.1142/S0219498804000897   DOI
19 P. Menal and J. Moncasi, On regular rings with stable range 2, J. Pure Appl. Algebra 24 (1982), no. 1, 25-40. https://doi.org/10.1016/0022-4049(82)90056-1   DOI
20 W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977), 269-278. https://doi.org/10.2307/1998510   DOI
21 A. C. Ozcan, A. Harmanci, and P. F. Smith, Duo modules, Glasg. Math. J. 48 (2006), no. 3, 533-545. https://doi.org/10.1017/S0017089506003260   DOI
22 Z.Wang, J. Chen, D. Khurana, and T.Y. Lam, Rings of idempotent stable range one, Algebr. Represent. Theory 15 (2012), no. 1, 195-200. https://doi.org/10.1007/s10468-011-9276-4   DOI
23 F. Perera, Lifting units modulo exchange ideals and C*-algebras with real rank zero, J. Reine Angew. Math. 522 (2000), 51-62. https://doi.org/10.1515/crll.2000.040
24 F. Siddique, On two questions of Nicholson, https://arxiv.org/pdf/1402.4706.pdf, (2014), 5 pages.
25 J. Ster, Lifting units in clean rings, J. Algebra 381 (2013), 200-208. https://doi.org/10.1016/j.jalgebra.2013.02.014   DOI
26 L. N. Vaserstein, The stable range of rings and the dimension of topological spaces, Funkcional. Anal. i Prilozen. 5 (1971), no. 2, 17-27.
27 L. N. Vaserstein, Bass's first stable range condition, J. Pure Appl. Algebra 34 (1984), no. 2-3, 319-330. https://doi.org/10.1016/0022-4049(84)90044-6   DOI
28 C. A. Weibel, The K-book, Graduate Studies in Mathematics, 145, American Mathematical Society, Providence, RI, 2013.
29 H.-P. Yu, On quasi-duo rings, Glasgow Math. J. 37 (1995), no. 1, 21-31. https://doi.org/10.1017/S0017089500030342   DOI
30 Y. Zhou, Generalizations of perfect, semiperfect, and semiregular rings, Algebra Colloq. 7 (2000), no. 3, 305-318. https://doi.org/10.1007/s10011-000-0305-9   DOI