# $PI$-extending modules via nontrivial complex bundles and Abelian endomorphism rings

Document Type : Research Paper

Authors

1 Hacettepe University‎, ‎Faculty of Science‎, ‎Department of Mathematics‎, ‎06532‎, ‎Beytepe‎, ‎Ankara‎, ‎Turkey.

2 Hacettepe University Department of Mathematics

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 paper, we focus on direct summands and indecomposable decompositions of $PI$-extending modules. To this end, we provide several counter examples including the tangent bundles of complex spheres of dimensions bigger than or equal to 5 and certain hyper surfaces in projective spaces over complex numbers and obtain results when the $PI$-extending property is inherited by direct summands. Moreover, we show that under some module theoretical conditions $PI$-extending modules with Abelian endomorphism rings have indecomposable decompositions. Finally, we apply our former results, getting that, under suitable hypotheses, the finite exchange property implies the full exchange property.

Keywords

Main Subjects

#### References

F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer, New York, 1992.
H. Bass, Algebraic K-Theory, Benjamin, New York, 1968.
G. F. Birkenmeier, A. Tercan and C. C. Yucel, The extending condition relative to sets of submodules, Comm. Algebra, 42 (2014), no. 2, 764--778.
N. V. Dung, D. Van Huynh, P. F. Smith and R. Wisbauer, Extending Modules, Longman, England, 1994.
T. J. Hodges and J. Osterburg, A rank two indecomposable projective module over a Noetherian domain of Krull dimension one, Bull. Lond. Math. Soc. 19 (1987), no. 2, 139--144.
T. J. Hodges and J. T. Stafford, Noetherian rings with big indecomposable projective modules, Bull. Lond. Math. Soc. 21 (1989), no. 3, 249--254.
F. Ischebeck and R. A. Rao, Ideals and Reality, Springer-Verlag, Berlin, Heidelberg, 2005.
Y. Kara and A. Tercan, Modules whose certain submodules are essentially embedded in direct summands, Rocky Mountain J. Math. 46 (2016), no. 2, 519--532.
L. S. Levy, Projectives of large uniform-rank in Krull dimension 1, Bull. Lond. Math. Soc. 21 (1989), no. 1, 57--64.
S. H. Mohamed and B. J. Muller, Continuous and Discrete Modules, London Math. Soc. Lecture Note Ser. 147, Cambridge Univ. Press, Cambridge, 1990.
M. P. Murthy, Zero cycles and projective modules, Math. Ann. (1994), no. 2, 405--434.
P. P. Nielsen, Square-free modules with the exchange property, J. Algebra 323 (2010), no. 7, 1993--2001.
P. F. Smith and A. Tercan, Continuous and quasi-continuous modules, Houston J. Math. 18 (1992), no. 3, 339--348.
P. F. Smith and A. Tercan, Direct summands of modules which satisfy (C11), Algebra Colloq. 11 (2004), no. 2, 231--237.
P. F. Smith and A. Tercan, Generalizations of CS-modules, Comm. Algebra 21 (1993), no. 6, 1809--1847.
B. Zimmermann-Huisgen and W. Zimmermann, Classes of modules with the exchange property, J. Algebra 88 (1984), no. 2, 416--434.

### History

• Receive Date: 05 November 2014
• Revise Date: 20 October 2015
• Accept Date: 20 October 2015