Vol. 30, No. 1, 1969

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Local domains with topologically T-nilpotent radical

James Reaves Smith

Vol. 30 (1969), No. 1, 233–245

This paper is concerned with local integral domains (no chain condition) which have the following property: for each ideal 𝒜0 of A and for each sequence (an)nN of elements of M(A), the maximal ideal of A, there is an M N such that a0a1ak ∈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.

Mathematical Subject Classification
Primary: 13.15
Received: 7 October 1968
Published: 1 July 1969
James Reaves Smith