A Note on Variation of Supplemented Modules

Bilhan G., Güroğlu A. T.

The Sixth China-Japan-Korea International Conference on Ring and Module Theory, Suweon, Güney Kore, 27 Haziran - 02 Temmuz 2011, (Özet Bildiri)

  • Yayın Türü: Bildiri / Özet Bildiri
  • Basıldığı Şehir: Suweon
  • Basıldığı Ülke: Güney Kore
  • Manisa Celal Bayar Üniversitesi Adresli: Evet

Özet

Let R be an associative ring with unity and M be an unital left R-module. A module M is called supplemented, if for every submodule A of M, there is a submodule B of M such that M = A + B and A \ B is a small submodule of B. A module M is amply supplemented, if whenever M = A + B, then B contains a supplement of A. We shall say that, a module M is w-supplemented, if every semisimple submodule of M has a supplement in M. A module M is called amply w-supplemented, if M = A + B where A is semisimple submodule of M, then B

contains a supplement of A. In this work, the properties of w-supplemented modules are studied and obtained the following some results.

Proposition : A module M is w-supplemented if and only if M is amply w- supplemented.

Lemma : An extension of w-supplemented module by w-supplemented is w-supplemented. That is, let M be a module and L be a submodule of M. If L and M=L are w-

supplemented and L ¿ M, then M is w-supplemented.

Proposition : Over a Dedekind domain R, all torsion modules are w-supplemented.

The following example shows that every submodule of w-supplemented module

need not be w-supplemented.