ON CLASSES OF MODULES CLOSED UNDER INJECTIVE HULLS AND ARTINIAN PRINCIPAL IDEAL RINGS

Abstract

In this work we consider some classes of modules closed under certain closure properties such as being closed under taking submodules, quotients, injective hulls and direct sums. We obtain some characterizations of artinian principal ideal rings using properties of big lattices of module classes.

Keywords

• Artinian principal ideal rings, big lattices of module classes, QF-rings

References

1. A. Alvarado, H. Rinc´on and J. R´ıos, On some lattices of module classes, J. Algebra Appl., 5(1) (2006), 105-117.
2. F. Anderson and K. Fuller, Rings and Categories of Modules, Springer-Verlag, New York, 1992.
3. A. Boyle, When projective covers and injective hulls are isomorphic, Bull. Aus- tral. Math. Soc., 8 (1973), 471-476.
4. R. Bronowitz and M. L. Teply, Torsion theories of simple type, J. Pure Appl. Algebra, 3 (1973), 329-336.
5. A. El Mejdani, H. Essannouni and A. Kaidi, Rings with nice Artinian Modules, Int. J. Algebra, 18(2) (2008), 895-904.
6. C. Faith, On K¨othe rings, Math. Ann., 164 (1966), 207-212.
7. C. Faith, Algebra II, Ring Theory, Grundlehren der mathematischen Wis- senschaften 191, Springer-Verlag, 1976.
8. F. Kasch, Modules and rings, Academic Press, London, 1982.

9. T. Y. Lam, Lectures on Modules and Rings, Springer- Verlag, New York, 1999.
10. A. Rosenberg and D. Zelinsky, Finiteness of the injective hull, Math. Z., 70 (1959), 372-378.
11. R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach Science Publishers, 1991.
12. Alejandro Alvarado-Garc´ıa, C´esar Cejudo-Castilla, Hugo Alberto Rinc´on-Mej´ıa, Ivan Fernando Vilchis-Montalvo Departmento de Matem´aticas Facultad de Ciencias Universidad Nacional Aut´onoma de M´exico Circuito Exterior, C.U. M´exico, D.F. e-mails: [e-mail] (A. Alvarado G.) [e-mail] (C. Cejudo C.) [e-mail] (H. A. Rinc´on M.) [e-mail] (I.F. Vilchis M.)