Projective modules over semilocal rings

Abstract

Let R be a commutative ring with a unit element.If there exist no proper ideals α, 6 such that R = a φ 5, then R is said to be indecomposable.If the number of maximal ideals of R is finite, then R is said to be semilocal.In [ 6 ], I. Kaplansky proved that, over a local ring, any projective module is free.Our objective in this paper is to generalize his theorem into THEOREM.Over a commutative indecomposable semilocal ring, any projective module is free.Every ring considered in this paper has a unit element which acts as unit operator on any module.Λ denotes a ring (not always commutative) and R denotes a commutative ring.Modules are always left modules.