On modules with chain condition on non-small submodules

In 1979, Fleury studied a class of modules with finite spanning dimension and dually a class of modules with ascending chain condition on non-small submodules was studied by Lomp and Ozcan in 2011. In the present work, we explore and investigate some new characterizations and properties of these classes of modules.

Keywords:

ns-Noetherian module, Noetherian module, spanning dimension small submodule, hollow module,

PDF

___

D. D. Anderson and M. Winders, Idealization of a module, J. Commut. Algebra, 1(1) (2009), 3-56.
N. V. Dung, D. V. Huynh, P. F. Smith and R. Wisbauer, Extending Modules, Pitman Research Notes in Mathematics Series, Vol. 313, Longman Scientific and Technical, Harlow, 1994.
P. Fleury, A note on dualizing Goldie dimension, Canad. Math. Bull., 17(4) (1974), 511-517.
A. Ghorbani and A. Haghany, Generalized Hopfian modules, J. Algebra, 255(2) (2002), 324-341.
K. R. Goodearl and R. B. Warfield, An Introduction to Noncommutative Noetherian Rings, London Mathematical Society Student Texts, 2nd edn., Cambridge University Press, Cambridge, 2004.
W. Heinzer, C. Rotthaus and S. Wiegand, Examples of non-Noetherian domains inside power series rings, J. Commut. Algebra, 6(1) (2014), 53-93.
D. Keskin, On lifting modules, Comm. Algebra, 28(7) (2000), 3427-3440.
T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics, Vol. 131, 2nd edn., Springer-Verlag, New York, 2001.
C. Lomp and A. C. Ozcan, Fleury's spanning dimension and chain conditions on non-essential elements in modular lattices, Colloq. Math., 124(1) (2011), 133-144.
S. H. Mohamed and B. J. Müller, Continuous and Discrete Modules, London Mathematical Society, Lecture Note Series, Vol. 147, Cambridge University Press, Cambridge, 1990.
K. M. Rangaswamy, Modules with finite spanning dimension, Canad. Math. Bull., 20(2) (1977), 255-262.
P. F. Smith and M. R. Vedadi, Modules with chain conditions on non-essential submodules, Comm. Algebra, 32(5) (2004), 1881-1894.
R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach Science Publishers, Philadelphia, PA, 1991.
O. Zariski and P. Samuel, Commutative Algebra, Vol. 1, Graduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg-Berlin, 1975.