Engin Büyükaşık
ON COFINITELY WEAK SUPPLEMENTED MODULES Engin Büyükaşık* Department of Mathematics, Izmir Institute of Technology, Gülbahçe Köyü, 35430, Urla, Izmir, Turkey
:اﻟﺨﻼﺻـﺔ وﻳﻜ ﻮن آ ﻞ ﻣﻌﻴ ﺎر ﻧ ﺼﻒ، اﻟﻤﺤﻠﻴﺔ ﺷﺮﻃًﺎ ﻣﻜﺎﻓﺌ ًﺎ ﻟﻜﻮﻧﻬ ﺎ ﻣ ﺼﺎﺣﺒﺔ ﻣﺤ ﺪودة ﺿ ﻌﻴﻔﺔ اﻻآﺘﻤ ﺎلnoetherian أن ﻟﻤﻌﺎﻳﻴﺮ- ﻓﻲ هﺬا اﻟﺒﺤﺚ- ﺳﻮف ﻧﺜﺒﺖ ﺿ ﻌﻴﻒ اﻻآﺘﻤ ﺎل إذا آ ﺎن آ ﻞ ﻣﻌﻴ ﺎر داﺋﺮﻳ ًﺎ ﺟﺰﺋﻴ ًﺎ ﺿ ﻌﻴﻒM وﺳﻨﺜﺒﺖ أن، ذات اﻷﺳﺎس اﻟﺼﻐﻴﺮM وﻟﻤﻌﻴﺎر، ﻣﺤﻠﻲ ﻣﺼﺎﺣﺒًﺎ ﻣﺤﺪودًا ﺿﻌﻴﻒ اﻻﺣﺘﻤﺎل . ﻣﻠﺘﻮﻳًﺎ ﻣﺼﺎﺣﺒًﺎ ﻣﺤﺪودًا ﺿﻌﻴﻒ اﻻآﺘﻤﺎلR– إذا آﺎن آﻞ ﻣﻌﻴﺎرh – ﻧﺼﻒ ﻣﺤﻠﻲR وﻳﻜﻮن اﻟﻨﻄﺎق اﻟﺘﺒﺎدﻟﻲ.اﻻآﺘﻤﺎل
ABSTRACT For a locally noetherian module, we prove some conditions equivalent to being cofinitely weak supplemented. Every semilocal module is cofinitely weak supplemented. For a module M with small radical, it is shown that M is weakly supplemented if and only if every cyclic submodule has a weak supplement. A commutative domain R is hsemilocal if and only if every torsion R-module is cofinitely weak supplemented. AMS 2000 Subject Classification: 13C05, 13C99, 16D10, 16P40 Key words: algebra, module theory
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[email protected] Paper Received: 15 March 2006; Accepted 4 May 2008
The Arabian Journal for Science and Engineering, Volume 34, Number 1A
January 2009
159
Engin Büyükaşık
ON COFINITELY WEAK SUPPLEMENTED MODULES
1. INTRODUCTION Throughout, we assume that R is an associative ring with identity and all modules are unital left R-modules. By R J(R) we denote the Jacobson radical of R. R is called semilocal if is semisimple. A commutative ring R is J (R) semilocal if and only if R has only finitely many maximal ideals, see [1, Proposition 20.2]. Let M be an R-module. A submodule L of M is small in M, denoted as L