Hostname: page-component-77c89778f8-n9wrp Total loading time: 0 Render date: 2024-07-21T18:05:29.328Z Has data issue: false hasContentIssue false
  • English
  • Français

Characterizations and Direct Sums of Unit-Endoregular Modules

Part of: Homological methods Modules, bimodules and ideals Rings and algebras arising under various constructions

Published online by Cambridge University Press:  10 August 2018

Xiaoxiang Zhang  and
Gangyong Lee
Xiaoxiang Zhang
Affiliation:
School of Mathematics, Southeast University Nanjing 211189, People's Republic of China ([email protected])
Gangyong Lee
Affiliation:
Department of Mathematics Education, Chungnam National University Daejeon 34134, Republic of Korea ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

A module is called unit-endoregular if its endomorphism ring is unit-regular. In this paper, we continue the research in unit-endoregular modules. More characterizations of unit-endoregular modules are obtained. As a special case, we show that for an abelian group G, Endℤ(G) is a unit-regular Baer ring if and only if Endℤ(G) is a two-sided extending regular ring. While the class of unit-endoregular modules is not closed under direct sums, we provide a characterization when there are direct sums of two or more unit-endoregular modules also unit-endoregular under certain conditions. In particular, we investigate unit-endoregular modules which are direct sums of indecomposable modules.

Keywords

unit-regular ringunit-endoregular modulestable range 1clean moduletwo-sided extending regular ring

MSC classification

Primary: 16D70: Structure and classification (except as in ), direct sum decomposition, cancellation
Secondary: 16S50: Endomorphism rings; matrix rings 16E50: von Neumann regular rings and generalizations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 4 , November 2018 , pp. 1103 - 1112
Copyright
Copyright © Edinburgh Mathematical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Azumaya, G., On generalized semi-primary rings and Krull–Remak–Schmidt's theorem, Jap. J. Math. 19 (1948), 525–547.Google Scholar
2.Camillo, V. P. and Khurana, D., A characterization of unit regular rings, Comm. Algebra 29(5) (2001), 2293–2295.Google Scholar
3.Camillo, V. P. and Yu, H. P., Exchange rings, units and idempotents, Comm. Algebra 22(12) (1994), 4737–4749.Google Scholar
4.Camillo, V. P., Khurana, D., Lam, T. Y., Nicholson, W. K. and Zhou, Y., Continuous modules are clean, J. Algebra 304(1) (2006), 94–111.Google Scholar
5.Canfell, M. J., Completion of diagrams by automorphisms and Bass’ first stable range condition, J. Algebra 176(2) (1995), 480–503.Google Scholar
6.Chatters, A. W. and Khuri, S. M., Endomorphism rings of modules over nonsingular CS rings, J. London Math. Soc. 21(2) (1980), 434–444.Google Scholar
7.Ehrlich, G., Unit-regular rings, Portugal. Math. 27 (1968), 209–212.Google Scholar
8.Ehrlich, G., Units and one-sided units in regular rings, Trans. Amer. Math. Soc. 216 (1976), 81–90.Google Scholar
9.Fuchs, L., On a substitution property of modules, Monatsh. Math. 75 (1971), 198–204.Google Scholar
10.Goodearl, K. R., Von Neumann regular rings (Pitman, London, 1979); 2nd edition, (Krieger, 1991).Google Scholar
11.Handelman, D., Perspectivity and cancellation in regular rings, J. Algebra 48(1) (1977), 1–16.Google Scholar
12.Henriksen, M., On a class of regular rings that are elementary divisor rings, Ark. Math. (Basel) 24(1) (1973), 133–141.Google Scholar
13.Krylov, P., Mikhalev, A. and Tuganbaev, A., Endomorphism rings of abelian groups, Algebra and Applications, Volume 2 (Kluwer Academic, Dordrecht 2003).Google Scholar
14.Lam, T. Y., A crash course on stable range, cancellation, substitution and exchange, J. Algebra Appl. 3(3) (2004), 301–343.Google Scholar
15.Lee, G., Rizvi, S. T. and Roman, C. S., Modules whose endomorphism rings are von Neumann regular, Comm. Algebra 41(11) (2013), 4066–4088.Google Scholar
16.Lee, G., Roman, C. S. and Zhang, X., Modules whose endomorphism rings are division rings, Comm. Algebra 42(12) (2014), 5205–5223.Google Scholar
17.Marks, G., A criterion for unit-regularity, Acta Math. Hungar. 111(4) (2006), 311–312.Google Scholar
18.Mohamed, S. H. and Müller, B. J., Continuous and discrete modules, Lecture Note Series, Volume 147 (London Mathematical Society, Cambridge University Press, 1990).Google Scholar
19.Nicholson, W. K. and Varadarajan, K., Countable linear transformations are clean, Proc. Amer. Math. Soc. 126(1) (1998), 61–64.Google Scholar
20.Rangaswamy, K. M., Abelian groups with endomorphic images of special types, J. Algebra 6 (1967), 271–280.Google Scholar
21.Rangaswamy, K. M., Representing Baer rings as endomorphism rings, Math. Ann. 190 (1970/1971), 167–176.Google Scholar
22.Rangaswamy, K. M., Regular and Baer rings, Proc. Amer. Math. Soc. 42 (1974), 354–358.Google Scholar
23.Stenström, B., Rings of quotients, Volume 217 (Springer, 1975).Google Scholar
24.Wang, Z., Chen, J., Khurana, D. and Lam, T. Y., Rings of idempotent stable range one, Algebr. Represent. Theory 15(1) (2012), 195–200.Google Scholar
25.Ware, R., Endomorphism rings of projective modules, Trans. Amer. Math. Soc. 155 (1971), 233–256.Google Scholar
26.Warfield, R. B. Jr., Cancellation of modules and groups and stable range of endomorphism rings, Pacific J. Math. 91(2) (1980), 457–485.Google Scholar
27.Wei, J. Q., Unit-regularity and stable range conditions, Comm. Algebra 33(6) (2005), 1937–1946.Google Scholar
28.Zhang, X. and Lee, G., Modules whose endomorphism rings are unit-regular, Comm. Algebra 44(2) (2016), 697–709.Google Scholar