on cofinitely weak supplemented modules

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

*E-mail: [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