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Dimensions like Gelfand, Krull, Goldie have an intrinsic role in the study oftheory of rings and modules. They provide useful technical tools for studyingtheir structure. In this paper we define one of the dimensions calledcouniserial dimension that measures how close a ring or module is to beinguniform. Despite their different objectives, it turns out that there arecertain common properties between the couniserial dimension and Krull dimensionlike each module having such a dimension contains a uniform submodule and hasfinite uniform dimension, among others. Like all dimensions, this is an ordinalvalued invariant. Every module of finite length has couniserial dimension andits value lies between the uniform dimension and the length of the module.Modules with countable couniserial dimension are shown to possessindecomposable decomposition. In particular, von Neumann regular ring withcountable couniserial dimension is semisimple artinian. If the maximal rightquotient ring of a non-singular ring R has a couniserial dimension as anR-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 Ris a semisimple artinian ring.

Indecomposable Decomposition and couniserial dimension