Rings for which every cyclic module is dual automorphism-invariant


Kosan M. T. , Nguyen Thi Thu Ha N. T. T. H. , Truong Cong Quynh T. C. Q.

JOURNAL OF ALGEBRA AND ITS APPLICATIONS, vol.15, no.5, 2016 (Peer-Reviewed Journal) identifier identifier

  • Publication Type: Article / Article
  • Volume: 15 Issue: 5
  • Publication Date: 2016
  • Doi Number: 10.1142/s021949881650078x
  • Journal Name: JOURNAL OF ALGEBRA AND ITS APPLICATIONS
  • Journal Indexes: Science Citation Index Expanded, Scopus
  • Keywords: (Dual) Automorphism-invariant module and ring, a-ring, q-ring, q*-ring, semiperfect ring

Abstract

Rings all of whose right ideals are automorphism-invariant are called right a-rings. In the present paper, we study rings having the property that every right cyclic module is dual automorphism-invariant. Such rings are called right a*-rings. We obtain some of the relationships a-rings and a*-rings. We also prove that; (i) A semiperfect ring R is a right a*-ring if and only if any right ideal in J(R) is a left T-module, where T is a subring of R generated by its units, (ii) R is semisimple artinian if and only if R is semiperfect and the matrix ring M-n(R) is a right a*-ring for all n > 1, (iii) Quasi-Frobenius right a*-rings are Frobenius.