INDEPENDENTLY GENERATED MODULES

Title & Authors
INDEPENDENTLY GENERATED MODULES
Kosan, Muhammet Tamer; Ozdin, Tufan;

Abstract
A module M over a ring R is said to satisfy (P) if every generating set of M contains an independent generating set. The following results are proved; (1) Let $\small{\tau}$ = ($\small{\mathbb{T}_\tau,\;\mathbb{F}_\tau}$) be a hereditary torsion theory such that $\small{\mathbb{T}_\tau}$ $\small{\neq}$ Mod-R. Then every $\small{\tau}$-torsionfree R-module satisfies (P) if and only if S = R/$\small{\tau}$(R) is a division ring. (2) Let $\small{\mathcal{K}}$ be a hereditary pre-torsion class of modules. Then every module in $\small{\mathcal{K}}$ satisfies (P) if and only if either $\small{\mathcal{K}}$ = {0} or S = R/$\small{Soc_\mathcal{K}}$(R) is a division ring, where $\small{Soc_\mathcal{K}}$(R) = $\small{\cap}${I $\small{4\leq}$ $\small{R_R}$ : R/I$\small{\in\mathcal{K}}$}.
Keywords
generated set for modules;basis;(non)-singular modules;division ring;torsion theory;
Language
English
Cited by
References
1.
F. W. Anderson and K. R. Fuller, Rings and Categories of Modules (Second edition), Graduate Texts in Mathematics, 13. Springer-Verlag, New York, 1992

2.
D. D. Anderson and J. Robeson, Bases for modules, Expo. Math. 22 (2004), no. 3, 283–296

3.
H. Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488

4.
J. Dauns and Y. Zhou, Classes of Modules, Pure and Applied Mathematics (Boca Raton), 281. Chapman & Hall/CRC, Boca Raton, FL, 2006

5.
J. S. Golan, Torsion Theories, Pitman Monographs and Surveys in Pure and Applied Mathematics, 29. Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1986

6.
K. R. Goodearl, Ring Theory: Nonsingular Rings and Modules, Pure and Applied Mathematics, No. 33. Marcel Dekker, Inc., New York-Basel, 1976

7.
J. Neggers, Cyclic rings, Rev. Un. Mat. Argentina 28 (1977), no. 2, 108–114

8.
W. H. Rant, Minimally generated modules, Canad. Math. Bull. 23 (1980), no. 1, 103– 105

9.
L. J. Ratliff and J. C. Robson, Minimal bases for modules, Houston J. Math. 4 (1978), no. 4, 593–596

10.
B. Stenstrom, Rings of Quotients, Springer-Verlag, 1975

11.
Y. Zhou, A characterization of left perfect rings, Canad. Math. Bull. 38 (1995), no. 3, 382–384

12.
Y. Zhou, Relative chain conditions and module classes, Comm. Algebra 25 (1997), no. 2, 543–557