Uniserial dimension of modules

Until now there has been no suitable dimension to measure how far a module deviates from being uniserial. We define and study a new dimension, which we call uniserial dimension. The uniserial dimension is a measure of how far a module deviates from being uniserial. It is shown that for a ring R and an ordinal number α, there exists an R-module of uniserial dimension α. We show that a commutative ring R is Noetherian (resp. Artinian) if and only if every finitely generated R-module has (resp. finite) uniserial dimension. We characterize rings whose modules have uniserial dimension. In fact, it is shown that every right R-module has uniserial dimension if and only if the free right R-module ⨁i=1∞R has uniserial dimension if and only if R is a semisimple Artinian ring.