Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T14:01:43.306Z Has data issue: false hasContentIssue false
  • English
  • Français

CORETRACTABLE MODULES

Part of: Modules, bimodules and ideals Local rings and generalizations

Published online by Cambridge University Press:  01 June 2009

B. AMINI ,
M. ERSHAD  and
H. SHARIF
B. AMINI*
Affiliation:
Department of Mathematics, College of Sciences, Shiraz University, Shiraz 71454, Iran (email: bamini@shirazu.ac.ir)
M. ERSHAD
Affiliation:
Department of Mathematics, College of Sciences, Shiraz University, Shiraz 71454, Iran (email: ershad@shirazu.ac.ir)
H. SHARIF
Affiliation:
Department of Mathematics, College of Sciences, Shiraz University, Shiraz 71454, Iran (email: sharif@susc.ac.ir)
*
✉For correspondence; e-mail: bamini@shirazu.ac.ir
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

An R-module M is called coretractable if  HomR(M/K,M)≠0 for any proper submodule K of M. In this paper we study coretractable modules and their endomorphism rings. It turns out that if all right R-modules are coretractable, then R is a right Kasch and (two-sided) perfect ring. However, the converse holds for commutative rings. Also, for a semi-injective coretractable module MR with S=EndR(M), we show that u.dim(SS)=corank(MR).

Keywords

coretractable moduleKasch ringperfect ringsemi-Artinian ringmax ringsemi-injective module

MSC classification

Secondary: 16D10: General module theory 16L30: Noncommutative local and semilocal rings, perfect rings
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 86 , Issue 3 , June 2009 , pp. 289 - 304
Copyright
Copyright © Australian Mathematical Society 2009

References

[1]Albu, T. and Wisbauer, R., ‘Kasch modules’, in: Advances in Ring Theory (eds. S. K. Jain and S. T. Rizvi) (Birkhäuser, Basel, 1997), pp. 1–16.Google Scholar
[2]Anderson, F. W. and Fuller, K. R., Rings and Categories of Modules (Springer, New York, 1992).CrossRefGoogle Scholar
[3]Fleury, P., ‘A note on dualizing Goldie dimension’, Canad. Math. Bull. 17 (1974), 511–517.CrossRefGoogle Scholar
[4]Haghany, A. and Vedadi, M. R., ‘Study of semi-projective retractable modules’, Algebra Colloq. 14 (2007), 489–496.CrossRefGoogle Scholar
[5]Khuri, S. M., ‘Endomorphism rings and lattice isomorphisms’, J. Algebra 56 (1979), 401–408.CrossRefGoogle Scholar
[6]Khuri, S. M., ‘Correspondence theorems for modules and their endomorphism rings’, J. Algebra 122 (1989), 380–396.CrossRefGoogle Scholar
[7]Lam, T. Y., A First Course in Noncommutative Rings (Springer, New York, 1991).CrossRefGoogle Scholar
[8]Lam, T. Y., Lectures on Modules and Rings (Springer, New York, 1999).CrossRefGoogle Scholar
[9]Smith, P. F., ‘Modules with many homomorphisms’, J. Pure Appl. Algebra 197 (2005), 305–321.Google Scholar
[10]Stenstrom, B., Rings of Quotients (Springer, New York, 1975).CrossRefGoogle Scholar
[11]Varadarajan, K., ‘Dual Goldie dimension’, Comm. Algebra 7 (1979), 565–610.CrossRefGoogle Scholar
[12]Wisbauer, R., Foundations of Module and Ring Theory (Gordon and Breach, Philadelphia, PA, 1991).Google Scholar
[13]Zelmanowitz, J. M., ‘Correspondences of closed submodules’, Proc. Amer. Math. Soc. 124 (1996), 2955–2960.CrossRefGoogle Scholar
[14]Zhou, Z. P., ‘A lattice isomorphism theorem for nonsingular retractable modules’, Canad. Math. Bull. 37 (1994), 140–144.Google Scholar