  Dimensions like Gelfand, Krull, Goldie have an intrinsic role in the study of
theory of rings and modules. They provide useful technical tools for studying
their structure. In this paper we define one of the dimensions called
couniserial dimension that measures how close a ring or module is to being
uniform. Despite their different objectives, it turns out that there are
certain common properties between the couniserial dimension and Krull dimension
like each module having such a dimension contains a uniform submodule and has
finite uniform dimension, among others. Like all dimensions, this is an ordinal
valued invariant. Every module of finite length has couniserial dimension and
its value lies between the uniform dimension and the length of the module.
Modules with countable couniserial dimension are shown to possess
indecomposable decomposition. In particular, von Neumann regular ring with
countable couniserial dimension is semisimple artinian. If the maximal right
quotient ring of a non-singular ring R has a couniserial dimension as an
R-module, then R is a semiprime right Goldie ring. As one of the applications,
it follows that all right R-modules have couniserial dimension if and only if R
is a semisimple artinian ring.
