Dimension on non-essential submodules.

In this paper, we introduce and study the concepts of non-essential Krull dimension and non-essential Noetherian dimension of an R -module, where R is an arbitrary associative ring. These dimensions are ordinal numbers and extend the notion of Krull dimension. They respectively rely on the behavior of descending and ascending chains of non-essential submodules. It is proved that each module with non-essential Krull dimension (respectively, non-essential Noetherian dimension) has finite Goldie dimension. We also show that a semiprime ring R with non-essential Noetherian dimension is uniform. [ABSTRACT FROM AUTHOR]