  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.
