Annihilator Condition on Modules

Mahdou N., Koç S., Yıldız Yılmaz E., Tekir Ü.

IRANIAN JOURNAL OF SCIENCE, cilt.47, ss.1713-1721, 2023 (SCI-Expanded)

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 47
  • Basım Tarihi: 2023
  • Dergi Adı: IRANIAN JOURNAL OF SCIENCE
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED)
  • Sayfa Sayıları: ss.1713-1721
  • Marmara Üniversitesi Adresli: Evet

Özet

Let R " role="presentation" >𝑅 be a commutative ring with 1≠0" role="presentation" >1≠0 and M " role="presentation" >𝑀 a unital R-module. M " role="presentation" >𝑀 is said to satisfy Property (A) if for each finitely generated ideal J " role="presentation" >𝐽 of R contained in ZR(M)" role="presentation" >𝑍𝑅(𝑀), there exists 0≠m∈M " role="presentation" >0≠𝑚∈𝑀 such that Jm=(0). " role="presentation" >𝐽𝑚=(0).  Also M " role="presentation" >𝑀 is said to satisfy Property (T) if for each finitely generated submodule N " role="presentation" >𝑁 of M " role="presentation" >𝑀 contained in T(M), " role="presentation" >𝑇(𝑀), there exists 0≠a∈R " role="presentation" >0≠𝑎∈𝑅 such that aN=(0). " role="presentation" >𝑎𝑁=(0). In this article, we study certain annihilator conditions on modules such as Property (A) " role="presentation" >(𝐴) and Property (T). " role="presentation" >(𝑇).  In addition to give many properties of modules satisfying Property (A) " role="presentation" >(𝐴) (Property (T)), " role="presentation" >(𝑇)), we characterize these classes of modules in terms of r-submodules and sr-submodules. Also, we give a method to construct non Noetherian rings in which every ideal satisfies Property (A).