Uniserial Dimensions of Finitely Generated Primary Modules Over A Discrete Valuation Domain

Recently, the notion of the uniserial dimension of a module over a commutative ring that measures of how far the module deviates from being uniserial was introduced by Nazemian in 2014. In this article we give some methods to determine the uniserial dimension of a finitely generated primary module over a discrete valuation domain. It is well known that finitely generated primary modules over a discrete valuation domain can be decomposed as a direct sum of finite cyclic submodules where the orders of the cyclic generators are called the elementary divisors of the module. We show that the uniserial dimension of the module is a function of the elementary divisors of the module.