Is a right perfect right self-injective ring right PF? (Q5930015)

Recall that a ring \(R\) is called right pseudo-Frobenius (PF) if \(R_R\) is an injective cogenerator, right Kasch if \(R_R\) contains a copy of each simple right \(R\)-module, left CS if every left ideal is essential in a direct summand of \(_RR\), and left principally injective (respectively left mininjective) if every \(R\)-homomorphism from a principal (resp. minimal) left ideal of \(R\) to \(R\) can be extended to \(R\). We denote the left singular ideal, left socle, right socle, and Jacobson radical of \(R\) by \(Z(_RR)\), \(\text{Soc}(_RR)\), \(\text{Soc}(R_R)\), and \(J(R)\) respectively. The main result of this paper states that if \(R\) is a semiperfect right self-injective ring in which \(Re\) contains a minimal left ideal for every primitive idempotent \(e\) then the following conditions are equivalent: (a) \(R\) is right PF; (b) \(R\) is left principally injective; (c) \(R\) is left mininjective; (d) \(J(R)=Z(_RR)\); (e) \(\text{Soc}(_RR)=\text{Soc}(R_R)\); (f) \(\text{Soc}(R_R)\) is an essential left ideal of \(R\); (g) every simple left \(R\)-module embeds in \(\text{Soc}(R_R)\). (As the author notes, the implications (a) \(\Rightarrow\) (b), (c), (d), (e), (f) and (g) appear in \textit{W. K. Nicholson} and \textit{M. F. Yousif} [J. Algebra 187, No. 2, 548-578 (1997; Zbl 0879.16002)].) He also proves that (i) a semiperfect left CS ring \(R\) is right Kasch if and only if \(\text{Soc}(R_R)\) is an essential left ideal of \(R\), (ii) a right perfect right self-injective ring \(R\) is right PF if and only if \(R/Z(_RR)\) is a semiprime ring, and (iii) a right perfect left quasi-continuous ring is left and right Kasch.