MODULES WHOSE MAXIMAL SUBMODULES ARE SUPPLEMENTS

Abstract

We study modules whose maximal submodules are supplements (direct summands). For a locally projective module, we prove that every maximal submodule is a direct summand if and only if it is semisimple and projective. We give a complete characterization of the modules whose maximal submodules are supplements over Dedekind domains

Keywords

• Locally projective module,Supplement submodule,2000 AMS Classification: 13 C 05

MODULES WHOSE MAXIMAL SUBMODULES ARE SUPPLEMENTS

Abstract

References

1. Alizade, R., Bilhan, G. and Smith, P. F. Modules whose maximal submodules have supple- ments, Comm. Algebra 29, 2389–2405, 2001.
2. Anderson, F. W. and Fuller, K. R. Rings and Categories of Modules (Springer, New York, ). Clark, J., Lomp, C., Vanaja, N. and Wisbauer, R. Lifting Modules. Supplements and Pro- jectivity in Module Theory, (Frontiers in Mathematics, Birkh¨auser, Basel, 2006).
3. Cohn, P. M. Basic Algebra (Springer, London, 2003).
4. Drinfeld, V. Infinite–dimensional vector bundles in algebraic geometry: an introduction, in The Unity of Mathematics(Birkh¨auser, Boston, 2006), 263–304.
5. Estrada, S., Guil Asensio, P. A., Prest, M. and Trlifaj, J. Model category structures arising from Drinfeld vector bundles, Work in progress. Dung, N. V., Huynh, D. V., Smith, P. F. and Wisbauer, R. Extending Modules (Long- man:Burnt Mill, 1994).
6. Eklof, P. C. Modules with strange decomposition properties. Infinite length modules, Biele- feld, 75–87, 1998 (Trends Math., Birkh¨auser, Basel, 2000).
7. Eklof, P. C. and Shelah, S. The Kaplansky test problems for ℵ1-separable groups, Proc. Amer. Math. Soc. 126 (7), 1901–1907, 1998.
8. Gruson, L. and Raynaud, M. Crit`eres de platitude et de projectivit´e. Techniques de “plati- fication” d’un module, Invent. Math. 13, 1–89, 1971.

9. Guil Asensio, P. A., Izurdiaga, M. C. and Torrecillas, B. Decomposition properties of strict Mittag-Leffler modules, J. Algebra 310, 290–302, 2007.
10. Guil Asensio, P. A., Izurdiaga, M. C. and Torrecillas, B. Accesible subcategories of modules and pathological objects, Forum Math. 22 (3), 485–507, 2010.
11. Kasch, F. Modules and Rings (Academic Press, London, 1982).
12. Lam, T. Y. Lectures on Modules and Rings (Springer-Verlag, New York, 1999).
13. Mcconnel, J. C. and Robson, J. C. Homomorphisms and extensions of modules over certain differential polynomial rings, J. Algebra 26, 319–342, 1973.
14. Mohamed, S. H. and M¨uller, B.J. Continuous and Discrete Modules (Cambridge University Press, New York, 1990).
15. Sharp, R. Y. Steps in Commutative Algebra (Cambridge University Press, Cambridge, 1990).
16. Sharpe, D. W. and Vamos, P. Injective Modules (Cambridge University Press, Cambridge, ). Wisbauer, R. Foundations of Modules and Rings (Gordon and Breach, Reading, 1991).
17. Zimmermann-Huisgen, B. Pure submodules of direct products of free modules, Math. Ann. (3), 233–245, 1976.
18. Zimmermann-Huisgen, B. On the abundance of ℵ1-separable modules, in: Abelian Groups and Noncommutative Rings, Contemp. Math.130, 167–180, 1992.
19. Z¨oschinger, H. Komplementierte moduln ¨uber Dedekindringen, J. Algebra 29, 42–56, 1974.