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Rings whose (proper) cyclic modules have cyclic automorphism-invariant hulls

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Abstract

The object of this article is associate to automorphism-invariant modules that are invariant under any automorphism of their injective hulls with cyclic modules and cyclic modules have cyclic automorphism-invariant hulls. The study of the first sequence allows us to characterize rings whose cyclic right modules are automorphism-invariant and to show that if R is a right Köthe ring, then R is an Artinian principal left ideal ring in case every cyclic right R-module is automorphism-invariant. The study of the second sequence leads us to consider a generalization of hypercyclic rings that are each cyclic R-module has a cyclic automorphism-invariant hull. Such rings are called right a-hypercyclic rings. It is shown that every right a-hypercyclic ring with Krull dimension is right Artinian.

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Acknowledgements

The authors are grateful to Professor A. Leroy for the organization: “NonCommutative rings and their Applications, VI LENS 24-27 June 2019”. The work of T. C. Quynh was supported in part by the Ministry of Education and Training (B2020-DNA-10).

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences, Gazi University, Ankara, Turkey

    M. Tamer Koşan

  2. Department of Mathematics, The University of Danang, University of Science and Education, 459 Ton Duc Thang, Danang city, Vietnam

    Truong Cong Quynh

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  1. M. Tamer Koşan
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  2. Truong Cong Quynh
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Correspondence to Truong Cong Quynh.

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Koşan, M.T., Quynh, T.C. Rings whose (proper) cyclic modules have cyclic automorphism-invariant hulls. AAECC 32, 385–397 (2021). https://doi.org/10.1007/s00200-021-00494-8

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  • DOI: https://doi.org/10.1007/s00200-021-00494-8

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