Hostname: page-component-745bb68f8f-lrblm Total loading time: 0 Render date: 2025-01-11T09:00:31.383Z Has data issue: false hasContentIssue false
  • English
  • Français

Modules which are invariant under monomorphisms of their injective hulls

Part of: Modules, bimodules and ideals

Published online by Cambridge University Press:  09 April 2009

A. Alahmadi ,
N. Er  and
S. K. Jain
A. Alahmadi
Affiliation:
Department of MathematicsOhio UniversityAthens, OH 45701USA e-mail: noyaner@yahoo.com
N. Er
Affiliation:
Department of MathematicsThe Ohio State University-NewarkOH 43055, USA
S. K. Jain
Affiliation:
Department of MathematicsThe Ohio State University-NewarkOH 43055, USA
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.

In this paper certain injectivity conditions in terms of extensions of monomorphisms are considered. In particular, it is proved that a ring R is a quasi-Frobenius ring if and only if every monomorphism from any essential right ideal of R into R(N)R can be extended to RR. Also, known results on pseudo-injective modules are extended. Dinh raised the question if a pseudo-injective CS module is quasi-injective. The following results are obtained: M is quasi-injective if and only if M is pseudo-injective and M2 is CS. Furthermore, if M is a direct sum of uniform modules, then M is quasi-injective if and only if M is pseudo-injective. As a consequence of this it is shown that over a right Noetherian ring R, quasi-injective modules are precisely pseudo-injective CS modules.

MSC classification

Secondary: 16D50: Injective modules, self-injective rings 16D70: Structure and classification (except as in ), direct sum decomposition, cancellation
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 79 , Issue 3 , December 2005 , pp. 349 - 360
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Alamelu, S., ‘On quasi-injective modules over Noetherian rings’, J. Indian Math. Soc. 39 (1975), 121–130.Google Scholar
[2]Dinh, H. Q., ‘A note on pseudo-injective modules’, Comm. Algebra 33 (2005), 361–369.CrossRefGoogle Scholar
[3]Dung, N. V., Huynh, D. V., Smith, P. F. and Wisbauer, R., Extending modules, Pitman Research Notes in Mathematics Ser. 313 (Pitman, 1994).Google Scholar
[4]Er, N., ‘Direct sums and summands of weak-CS modules and continuous modules’, Rocky Mountain J. Math. 29 (1999), 491–503.CrossRefGoogle Scholar
[5]Er, N., ‘When submodules isomorphic to complements are complements’, East-West J. Math. 4 (2002), 1–12.Google Scholar
[6]Goodearl, K. R., Singular torsion and the splitting properties, Amer. Math. Soc. Mem. 124 (Amer. Math. Soc., Providence, RI, 1972).CrossRefGoogle Scholar
[7]Huynh, D. V., ‘A right countably sigma-CS ring with ACC or DCC on projective principal right ideals is left artnian and QF-3’, Trans. Amer. Math. Soc. 347 (1995), 3131–3139.CrossRefGoogle Scholar
[8]Jain, S. K. and Singh, S., ‘On pseudo injective modules and self pseudo injective rings’, J. Math. Sci. 2 (1967), 23–31.Google Scholar
[9]Jain, S. K. and Singh, S., ‘Quasi-injective and pseudo-injective modules’, Canad. Math. Bull. 18 (1975), 359–366.CrossRefGoogle Scholar
[10]Mohamed, S. H. and Müller, B. J., continuous and Discrete Modules (Cambridge University Press, Cambridge, 1990).CrossRefGoogle Scholar
[11]Okado, M., ‘On the decomposition of extending modules’, Math. Japonica 29 (1984), 939–941.Google Scholar
[12]Osofsky, B. L., ‘Rings all of whose finitely generated modules are injective’, Pacific J. Math. 14 (1964), 646–650.CrossRefGoogle Scholar
[13]Singh, S., ‘On pseudo-injective modules’, Riv. Mat. Univ. Parma (2) 9 (1968), 59–65.Google Scholar
[14]Teply, M. L., ‘Pseudo-injective modules which are not quasi-injective’, Proc. Amer. Math. Soc. 49 (1975), 305–310.CrossRefGoogle Scholar
[15]Tuganbaev, A. A., ‘Pseudo-injective modules and automorphisms’, Trudy Sem. Petrovsk. 4 (1978), 241–248.Google Scholar
[16]Zhongkui, L., ‘On X-extending and X-continuous modules’, Comm. Algebra 29 (2001), 2407–2418.CrossRefGoogle Scholar