Local duo rings whose finitely generated modules are direct sums of cyclics

Abstract

In this paper, we give an answer to the following question of Kaplansky [14] in the local case: For which duo rings R is it true that every finitely generated left R-module can be decomposed as a direct sum of cyclic modules? More precisely, we prove that for a local duo ring R, the following are equivalent: (i) Every finitely generated left R-module is a direct sum of cyclic modules; (ii) Every 2-generated left R-module is a direct sum of cyclic modules; (iii) Every factor module of R R ⊕ R is a direct sum of cyclic modules; (iv) Every factor module of R R ⊕ R is serial; (v) Every finitely generated left R-module is serial; (vi) R is uniserial and for every non-zero ideal I of R, R/I is a linearly compact left R-module; (vii) R is uniserial and every indecomposable injective left R-module is left uniserial; and, (viii) Every finitely generated right R-module is a direct sum of cyclic modules.