On semiperfect $a$-rings - Van, Truong Thi Thuy; Alghamdi, Ahmad M.; Alkinani, Amnah Abdu (Ukrains’kyi Matematychnyi Zhurnal, 2024)
UDC 512.5A 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.