SOME PROPERTIES OF EF-EXTENDING RINGS

In (16), Thuyet and Wisbauer considered the extending property for the class of (essentially) finitely generated submodules. A module M is called ef-extending if every closed submodule which con- tains essentially a finitely generated submodule is a direct summand of M. A ring R is called right ef-extending if RR is an ef-extending mod- ule. We show that a ring R is right ef-extending and the R-dual of every simple left R-module is simple if and only if R is semiperfect right con- tinuous with Sl = Sle RR. We also prove that a ring R is a QF-ring if and only if R is left Kasch and R (!) R is ef-extending if and only if R is right AGP-injective satisfying DCC on right (or left) annihilators and (RR)R is ef-extending. Throughout the paper, R represents an associative ring with identity 1 6 0 and all modules are unitary R-modules. We write MR (resp., RM) to indicate that M is a right (resp., left) R-module. We also write J (resp., Zr, Sr) for the Jacobson radical (resp., the right singular ideal, the right socle of R) and E(MR) for the injective hull of MR. If X is a subset of R, the right (resp., left) annihilator of X in R is denoted by rR(X) (resp., lR(X)) or simply r(X) (resp., l(X)) if no confusion appears. If N is a submodule of M (resp., proper submodule) we denote by N ≤ M (resp., N < M). Moreover, we write N ≤ e M, N ≤ ⊕ M and N ≤ max M to indicate that N is an essential submodule, a direct summand and a maximal submodule of M, respectively. A module M is called uniform if M 6 0 and every non-zero submodule of M is essential in M. It is called that a module M has finite uniform dimension if M does not contain an infinite direct sum of non-zero submodules. Let M,N be R-modules. M is said to be N-injective if, for any submodule H of N, every R-homomorphism f : H −→ M can be extended to an R-homomorphism ¯