[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.5666/KMJ.2019.59.3.391

The π-extending Property via Singular Quotient Submodules

Kara, Yeliz (Department of Mathematics, Bursa Uludag University)
Tercan, Adnan (Department of Mathematics, Hacettepe University)

Publication Information

Kyungpook Mathematical Journal / v.59, no.3, 2019 , pp. 391-401 More about this Journal

Abstract

A module is said to be ${\pi}$ -extending provided that every projection invariant submodule is essential in a direct summand of the module. In this article, we focus on the class of modules having the ${\pi}$ -extending property by looking at the singularity of quotient submodules. By doing so, we provide counterexamples, using hypersurfaces in projective spaces over complex numbers, to show that being generalized ${\pi}$ -extending is not inherited by direct summands. Moreover, it is shown that the direct sums of generalized ${\pi}$ -extending modules are generalized ${\pi}$ -extending.

Keywords

extending modules; ${\pi}$ -extending module; projection invariant;

Citations & Related Records

Reference

1	G. F. Birkenmeier, J. Y. Kim, J. K. Park, On polynomial extensions of principally quasi-baer rings, Kyungpook Math. J., 40(2000), 247-253.
2	G. F. Birkenmeier, A. Tercan and C. C. Yucel, The extending condition relative to sets of submodules, Comm. Algebra, 42(2014), 764-778. DOI
3	G. F. Birkenmeier, J. K. Park and T. Rizvi, Extensions of rings and modules, Birkhauser, New York, 2013.
4	G. F. Birkenmeier, Y. Kara and A. Tercan, ${\pi}$ -Baer rings, J. Algebra Appl., 17(2)(2018), 1850029, 19 pp.
5	N. V. Dung, D. V. Huynh, P. F. Smith and R. Wisbauer, Extending modules, Pitman, London, 1994.
6	L. Fuchs, Infinite abelian groups I, Academic Press, New York, 1970.
7	K. R. Goodearl, Ring theory: Nonsingular rings and modules, Marcel Dekker, New York, 1976.
8	I. Kaplansky, Infinite abelian groups, University of Michigan Press, 1969.
9	T. Y. Lam, Lectures on modules and rings, Springer-Verlag, Berlin, 1999.
10	Y. Kara, A. Tercan and R. Yasar, PI-extending modules via nontrivial complex bundles and abelian endomorphism rings, Bull. Iranian Math. Soc., 43(1)(2017), 121-129.
11	S. H. Mohamed and B. J. Muller, Continuous and discrete modules, London Mathematical Society Lecture Note Series 147, Cambridge University Press, Cambridge, 1990.
12	M. P. Murthy, Zero cycles and projective modules, Ann. Math., 140(1994), 405-434. DOI
13	P. F. Smith and A. Tercan, Generalizations of CS-modules, Comm. Algebra, 21(1993), 1809-1847. DOI
14	P. F. Smith and A. Tercan, Direct summands of modules which satisfy $C_{11}$ , Algebra Colloq., 21(2004), 231-237. DOI
15	A. Tercan and C. C. Yucel, Module theory, Extending modules and generalizations, Birkhauser, Basel, 2016.