It is proved here that a ring $R$ is right pseudo-Frobenius if and only if $R$ is a right Kasch ring such that the second right singular ideal is injective.

It is shown that if R is a right Noetherian ring whose right socle is essential as a right ideal and is contained in the left socle, then R is right Artinian. This result may be viewed as a one-sided version of a result of Ginn and Moss on two-sided Noetherian rings with essential socle. This also extends the work of Nicholson and Yousif where the same result is obtained under a stronger hypothesis. We use our work to obtain partial positive answers to some open questions on right CF, right FGF and right Johns rings.

We show that if R is a ring such that each minimal left ideal is essential in a (direct) summand of RR, then the dual of each simple right R-module is simple if and only if R is semiperfect with Soc(RR)=Soc(RR) and Soc(Re) is simple and essential for every local idempotent e of R. We also show that R is left CS and right Kasch if and only if R is a semiperfect left continuous ring with Soc(RR)⊆eRR. As a particular case of both results we obtain that R is a ring such that every (essential) closure of a minimal left ideal is summand (R is then said to be left strongly min-CS) and the dual of each simple right R-module is simple if and only if R is a semiperfect left continuous ring with Soc(RR)=Soc(RR)⊆eRR. Moreover, in this case R is also left Kasch, Soc(eR)≠0 for every local idempotent e of R, and R admits a (Nakayama) permutation of a basic set of primitive idempotents. As a consequence of this result we characterise left PF rings in terms of simple modules over the 2×2 matrix ring by showing that R is left PF if and only if M2(R) is a left strongly min-CS ring such that the dual of every simple right module is simple.

It is shown that a two-sided continuous ring with ascending chain condition on left annihilators is quasi-Frobenius.

Boyle and Goodearl proved that if R is a left V-ring then R has left Krull dimension if and only if R is left Neotherian. In this paper we extend this result to arbitrary V-modules.