Generalizations of coatomic modules

Open Mathematics (2005)

• Volume: 3, Issue: 2, page 273-281
• ISSN: 2391-5455

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Abstract

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For a ring R and a right R-module M, a submodule N of M is said to be δ-small in M if, whenever N+X=M with M/X singular, we have X=M. Let ℘ be the class of all singular simple modules. Then δ(M)=Σ{ L≤ M| L is a δ-small submodule of M} = Re jm(℘)=∩{ N⊂ M: M/N∈℘. We call M δ-coatomic module whenever N≤ M and M/N=δ(M/N) then M/N=0. And R is called right (left) δ-coatomic ring if the right (left) R-module R R(RR) is δ-coatomic. In this note, we study δ-coatomic modules and ring. We prove M=⊕i=1n Mi is δ-coatomic if and only if each M i (i=1,…, n) is δ-coatomic.

How to cite

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M. Koşan, and Abdullah Harmanci. "Generalizations of coatomic modules." Open Mathematics 3.2 (2005): 273-281. <http://eudml.org/doc/268730>.

@article{M2005,
abstract = {For a ring R and a right R-module M, a submodule N of M is said to be δ-small in M if, whenever N+X=M with M/X singular, we have X=M. Let ℘ be the class of all singular simple modules. Then δ(M)=Σ\{ L≤ M| L is a δ-small submodule of M\} = Re jm(℘)=∩\{ N⊂ M: M/N∈℘. We call M δ-coatomic module whenever N≤ M and M/N=δ(M/N) then M/N=0. And R is called right (left) δ-coatomic ring if the right (left) R-module R R(RR) is δ-coatomic. In this note, we study δ-coatomic modules and ring. We prove M=⊕i=1n Mi is δ-coatomic if and only if each M i (i=1,…, n) is δ-coatomic.},
author = {M. Koşan, Abdullah Harmanci},
journal = {Open Mathematics},
keywords = {16D60; 16D99; 16S90},
language = {eng},
number = {2},
pages = {273-281},
title = {Generalizations of coatomic modules},
url = {http://eudml.org/doc/268730},
volume = {3},
year = {2005},
}

TY - JOUR
AU - M. Koşan
AU - Abdullah Harmanci
TI - Generalizations of coatomic modules
JO - Open Mathematics
PY - 2005
VL - 3
IS - 2
SP - 273
EP - 281
AB - For a ring R and a right R-module M, a submodule N of M is said to be δ-small in M if, whenever N+X=M with M/X singular, we have X=M. Let ℘ be the class of all singular simple modules. Then δ(M)=Σ{ L≤ M| L is a δ-small submodule of M} = Re jm(℘)=∩{ N⊂ M: M/N∈℘. We call M δ-coatomic module whenever N≤ M and M/N=δ(M/N) then M/N=0. And R is called right (left) δ-coatomic ring if the right (left) R-module R R(RR) is δ-coatomic. In this note, we study δ-coatomic modules and ring. We prove M=⊕i=1n Mi is δ-coatomic if and only if each M i (i=1,…, n) is δ-coatomic.
LA - eng
KW - 16D60; 16D99; 16S90
UR - http://eudml.org/doc/268730
ER -

References

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1. [1] F.W. Anderson and K.R. Fuller: Rings and Categories of Modules, Springer-Verlag, New York, 1974. Zbl0301.16001
2. [2] K.R. Goodearl: Ring Theory: Nonsingular Rings and Modules, Dekker, New York, 1976.
3. [3] G. Gungoroglu: “Coatomic Modules”, Far East J. Math. Sci., Special Volume, Part II, (1998), pp. 153–162. Zbl1114.16300
4. [4] F. Kasch: Modules and Rings, Academic Press, 1982.
5. [5] C. Lomp: “On Semilocal Modules and Rings”, Comm. Alg., 27(4), (1999), pp. 1921–1935. Zbl0927.16012
6. [6] S.H. Mohamed and B.J. Müller: Continuous and discrete modules, London Math. Soc. LNS 147, Cambridge Univ. Press, Cambridge, 1990.
7. [7] R. Wisbauer: Foundations of Module and Ring Theory, Gordon and Breach, Reading, 1991.
8. [8] Y. Zhou: “Generalizations of Perfect, Semiperfect and Semiregular Rings”, Algebra Colloquium, Vol. 7(3), (2000), pp. 305–318. http://dx.doi.org/10.1007/s10011-000-0305-9 Zbl0994.16016
9. [9] M.Y. Yousif and Y. Zhou: “Semiregular, Semiperfect and Perfect Rings relative to an ideal”, Rocky Mountain J. Math., Vol. 32(4), (2002), pp. 1651–1671. Zbl1046.16007

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