On semiperfect $a$-rings

Truong Thi Thuy Van; Ahmad M.  Alghamdi; Amnah Abdu Alkinani

doi:10.3842/umzh.v76i5.7491

Truong Thi Thuy Van Faculty of Basic Sciences, Vinh Long University of Technology Education, Nguyen Hue, Vinh Long, Vietnam
Ahmad M. Alghamdi Mathematics Department, Faculty of Sciences, Umm Al-Qura University, Makkah, Saudi Arabia
Amnah Abdu Alkinani Department of Basic Science, Adham University College, Umm Al-Qura University, Makkah, Saudi Arabia

DOI: https://doi.org/10.3842/umzh.v76i5.7491

Keywords: semiperfect ring, Nakayama permutation, automorphism-invariant module

Abstract

UDC 512.5

A ring is called a right $a$-ring if every right ideal is automorphism invariant. We describe some properties of $a$-rings over semiperfect rings. It is shown that an I-finite right $a$-ring is a direct sum of a semisimple Artinian ring and a basic ring. It is also demonstrated that if $R$ is an indecomposable (as a ring) I-finite right $a$-ring not simple with nontrivial idempotents such that every minimal right ideal is a right annihilator and ${\rm Soc}(R_R)={\rm Soc}(_RR)$ is essential in $R_R$, then $R$ is a quasi-Frobenius ring and it is also a right $q$-ring.

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