LOCAL DUO RINGS WHOSE FINITELY GENERATED MODULES ARE DIRECT SUMS OF CYCLICS

摘要

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 circle plus R is a direct sum of cyclic modules; (iv) Every factor module of R-R circle plus 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.