Distributive rings, uniserial rings of fractions, and endo-Bezout modules. (Q1400880)

Let \(R\) be an associative ring with identity element. A right \(R\)-module \(M\) is said to be `distributive' (resp. `uniserial') if its lattice of submodules is distributive (resp. a chain). A distributive (resp. uniserial) ring is a ring \(R\) such that both modules \(_RR\) and \(R_R\) are distributive (resp. uniserial). A module is said to be `Bezout' if all its finitely generated submodules are cyclic. This paper consists of three sections. In the first one the author investigates localization of distributive rings. The second section studies orders in uniserial rings, and the last one investigates modules over endomorphism rings. Thus, among many other results, the author proves that a right \(R\)-module \(M\) is distributive if and only if for any quasi-injective right \(R\)-module \(N\), the left \(\text{End}(N)\)-module \(\Hom_R(M,N)\) is a Bezout module.