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http://dx.doi.org/10.5666/KMJ.2010.50.1.101

When Some Complement of an EC-Submodule is a Direct Summand  

Denizli, Canan Celep Yucel (Department of Mathematics, Faculty of Science and Art, Pamukkale University)
Ankara, Adnan Tercan (Department of Mathematics, Hacettepe University)
Publication Information
Kyungpook Mathematical Journal / v.50, no.1, 2010 , pp. 101-107 More about this Journal
Abstract
A module M is said to satisfy the $EC_{11}$ condition if every ec-submodule of M has a complement which is a direct summand. We show that for a multiplication module over a commutative ring the $EC_{11}$ and P-extending conditions are equivalent. It is shown that the $EC_{11}$ property is not inherited by direct summands. Moreover, we prove that if M is an $EC_{11}$-module where SocM is an ec-submodule, then it is a direct sum of a module with essential socle and a module with zero socle. An example is given to show that the reverse of the last result does not hold.
Keywords
Extending module; ec-closed submodule; P-extending module; $C_{11}$-module; Multiplication module;
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