A Note on the Existence of Indecomposable Essential Submodules of the Ring of Quotients of Ore Domains

TL;DR

This paper investigates the existence and bounds of indecomposable essential submodules within direct sums of the ring of quotients of Ore domains, providing theoretical bounds and illustrative examples.

Contribution

It establishes lower bounds for the size of direct sums containing indecomposable essential submodules in Ore domains with multiple irreducibles, advancing understanding of module structure.

Findings

Lower bounds for the supremum of cardinals for indecomposable essential submodules

Bounds on the number of times these bounds are attained

Examples illustrating the theoretical results

Abstract

We study the problem of existence of essential indecomposable submodules of direct sums of copies of the ring of quotients of Ore domains. We provide, for each Ore domain $D$, with at least three non-associate irreducibles, a lower bound for the supremum of all cardinals $\kappa$ such that the direct sum of $\kappa$ copies of the ring of quotients of $D$ contains indecomposable essential $D$-submodules, and a lower bound for the number of times this bound is attained up to isomorphisms. We provide examples illustrating these results.