r-Submodules and uz-modules

Document Type : Research Paper

Author

Department of Mathematics, Faculty of Mathematical Sciences and Computer, Shahid Chamran University of Ahvaz, Ahvaz, Iran.

Abstract

In this article we study and investigate the behavior of r-submodules (a proper submodule N of an R-module M  in which amN with AnnM(a)=(0) implies that mN for each aR and mM). We show that every simple submodule, direct summand,  divisible submodule, torsion submodule and the socle of a module is an r-submodule and if R is a domain, then the singular submodule is an r-submodule. We also introduce the concepts of uz-module (i.e., an R-module M such that either AnnM(a)(0) or aM=M, for every aR) and strongly  uz-module (i.e., an R-module M such that aMa2M, for every aR) in the category of modules over commutative rings. We show that every Von Neumann regular module is a strongly uz-module and every Artinian  R-module is a   uz-module. It is observed that if M is a faithful cyclic R-module, then  M is a   uz-module if and only if every its cyclic submodule is an r-submodule. In addition, in this case, R is a domain if and only if the only r-submodule of M is zero submodule. Finally, we prove that R  is a uz-ring if and only if every faithful cyclic R-module is a uz-module.

Keywords


[1] F. W. Anderson, K. R. Fuller, Rings and Categories of Modules, Graduate Texts in Mathematics, Vol. 13, 2nd Ed., Springer-Verlag, Berlin/Heidelberg, New York, 1992.
[2] M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesely, Reading Mass,1969.
[3] R. Gilmer, Multiplicative Ideal Theory, Marcel Dekker Inc, New York, 1972.
[4] R. Gilmer, W. Heinzer, On Jonsson modules over a commutative ring, Acta Sci. Math. 46 (1983), 3-15.
[5] J. A. Huckabo, Commutative Ring with Zero Divisors, Marcel Dekker Inc, 1988.
[6] T. W. Hungerford, Algebra, Graduate Texts in Mathematics, Vol. 73, Springer-Verlag, Berlin/Heidelberg,New York, 1989.
[7] I. Kaplansky, Commutative Rings, Allyn and Bacon, Boston, 1970.
[8] O. A. S. Karamzadeh, Module whose countably generated submodules are epimorphic images, Colloq. Math. 46 (1982), 143-146.
[9] S. Koc, U. Tekir, r-Submodules and special r-Submodules, Turk. J. Math. 42 (2018), 1863-1876.
[10] T. Y. Lam, Lecture on Modules and Rings, Springer, 1999.
[11] J. Lambek, Lecture on Rings and Modules, Waltham-Toronto-London: Blaisdell, 1966.
[12] R. Mohamadian, r-ideals in commutative rings, Turk. J. Math. 39 (2015), 733-749.
[13] R. Y. Sharp, Steps in Commutative Algebra, Cambridge university press, 1990.