A new dimension for Grothendieck categories via the atom spectrum

Abstract

In this paper, we define a new dimension for objects in a Grothendieck category $\mathcal{A}$. We show that it serves as a lower bound for Gabriel-Krull dimension and under certain conditions, the two dimensions coincide. We carry out our investigation for a fully right bounded ring $A$. We introduce a new spectrum Comp$\,A$ via compressible right $A$-modules. In analogy with dimension theory for commutative rings, we show that the Krull dimension of right $A$-modules can be computed via the length of chain of prime ideals of $A$ and also the length of chain of elements of Comp$\,A$.