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In this paper modules with ascending chain condition (acc)(respectively, with descending chain conditon (dcc)) on essential submodules are investigated: Finitely generated quasi-injective modules with this property are noetherian (resp., artinian). Also, it is shown 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 (yon Neumann) regular, left self-injective ring. Finally we obtain a new characterization of quasi-Frobenius rings (briefly, QF-rings): A left self-injective ring with acc on essential left or right ideals is a QF-ring.

1. All rings R considered here are associative with identity and all modules are unitary. Let M be an R-module. Then the socle of M is denoted by Soc (M). A submodule N of M is essential in M if for each nonzero submodule L of M, L c~ N:# 0. M has finite Goldie dimension if M does not contain an infinite direct sum of non-zero submodules. A module M is called quasi-injective (resp., quasi-projective) if M is M-injective (resp., M-projective)(see [1, w For a subset A of a ring R, r (A) and l_ (A) denote the right and left annihilators of A in R, respectively.