This paper is concerned
with local integral domains (no chain condition) which have the following
property: for each ideal 𝒜≠0 of A and for each sequence (an)n∈N of elements of
M(A), the maximal ideal of A, there is an M ∈ N such that a0a1⋯ak∈𝒜. A
local domain with this property is called a local domain with TTN. These
rings are shown to be rings with Krull dimension 1 and local domains with
Krull dimension 1 are shown to be dominated by rank 1 valuation rings.
Modules over these rings are studied and results concerning divisibility and
existence of simple submodules are obtained. Noetherian integral domains with
TTN are studied. Integral extensions of these rings are also studied. By
localization of previous results, a characterization is given of those integral
domains A wilh the property that every nonzero torsion A-module has a simple
submodule.
