Abstract
In this paper, we introduce and study the concepts of non-essential Krull dimension and non-essential Noetherian dimension of an [Formula: see text]-module, where [Formula: see text] 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 [Formula: see text] with non-essential Noetherian dimension is uniform.
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