• 제목/요약/키워드: summand sum property

검색결과 4건 처리시간 0.017초

DIRECT PROJECTIVE MODULES WITH THE SUMMAND SUM PROPERTY

  • Han, Chang-Woo;Choi, Su-Jeong
    • 대한수학회논문집
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    • 제12권4호
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    • pp.865-868
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    • 1997
  • Let R be a ring with a unity and let M be a unitary left R-module. In this paper, we establish [5, Proposition 2.8] by showing the proof of it. Moreover, from the above result, we obtain some properties of direct projective modules which have the summand sum property.

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SOME PROPERTIES OF A DIRECT INJECTIVE MODULE

  • Chang, Woo-Han;Choi, Su-Jeong
    • 한국수학교육학회지시리즈B:순수및응용수학
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    • 제6권1호
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    • pp.9-12
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    • 1999
  • The purpose of this paper is to show that by the divisibility of a direct injective module, we obtain some results with respect to a direct injective module.

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On Generalizations of Extending Modules

  • Karabacak, Fatih
    • Kyungpook Mathematical Journal
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    • 제49권3호
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    • pp.557-562
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    • 2009
  • A module M is said to be SIP-extending if the intersection of every pair of direct summands is essential in a direct summand of M. SIP-extending modules are a proper generalization of both SIP-modules and extending modules. Every direct summand of an SIP-module is an SIP-module just as a direct summand of an extending module is extending. While it is known that a direct sum of SIP-extending modules is not necessarily SIP-extending, the question about direct summands of an SIP-extending module to be SIP-extending remains open. In this study, we show that a direct summand of an SIP-extending module inherits this property under some conditions. Some related results are included about $C_{11}$ and SIP-modules.

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

  • Denizli, Canan Celep Yucel;Ankara, Adnan Tercan
    • Kyungpook Mathematical Journal
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    • 제50권1호
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    • pp.101-107
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    • 2010
  • 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.