Quasi-injective modules with acc or dcc on essential submodules (Q1111666)

It is shown that finitely generated modules with the properties given in the title are noetherian resp. artinian. Another result in the paper is that the endomorphism ring of a quasi-injective, quasi-projective module with acc on essential submodules is a ring direct sum of a left artinian ring and a von Neumann regular, left self-injective ring. Also, a new characterization of QF-rings is given: A left self-injective ring with acc on essential left or right ideals is a QF-ring. Recall that [in \textit{E. Armendariz}, Commun. Algebra 8, 299-308 (1980; Zbl 0444.16015)], QF- rings are characterized as left self-injective rings with dcc on essential left or right ideals.