Dimension of uncountably generated submodules

Abstract

In this article we introduce and study the concepts of uncountably generated Krull dimension and uncountably generated 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 and Noetherian dimension. They respectively rely on the behavior of descending and ascending chains of uncountably generated submodules. It is proved that a quotient finite dimensional module $M$ has uncountably generated Krull dimension if and only if it has Krull dimension, but the values of these dimensions might differ. Similarly, a quotient finite dimensional module $M$ has uncountably generated Noetherian dimension if and only if it has Noetherian dimension. We also show that the Noetherian dimension of a quotient finite dimensional module $M$ with uncountably generated Noetherian dimension $\beta$ is less than or equal to $\omega _{1}+\beta $, where $\omega_{1}$ is the first uncountable ordinal number.