Local Dimension of Coatomic Modules

We define and study local dimension for coatomic modules. Local dimension is a measure of how far a coatomic module deviates from being local. Every Noetherian module has local dimension. It is shown that a ring R with finite local dimension is semilocal. We study rings over which modules are coatomic and have local dimension. We show that, for a ring R, every right R-module is coatomic and has local dimension if and only if the free right R-module is coatomic and has local dimension, if and only if R is a semisimple Artinan ring. We obtain a characterization of right Artinian rings as those right Noetherian rings over which every finitely generated right module has finite local dimension. We show that a commutative ring R has (resp. finite) local dimension if and only if R is either Noetherian (resp. Artinian) or local.