Vol. 36, No. 3, 1971

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Topologies for quotient fields of commutative integral domains

Joby Milo Anthony

Vol. 36 (1971), No. 3, 585–601

In this paper topologies for the quotient field K of a commutative inSegral domain A are investigated. The topologies for K are defined so that convergence in K is stronger than convergence in A whenever A is a topological ring. In particular, the Mikusinski field of operators is the quotient field of many commutative integral domains which are also topological rings. Each of these rings leads to a topological convergence notion in the Mikusinski field, which is stronger than the convergence notion introduced originally by Mikusinski. (The latter has recently been shown to be nontopological.) In general, the algebraic and topological structures considered are not necessarily compatible; however, the question of compatibility is investigated. Necessary and sufficient conditions are given for the topology on A to be the restriction to A of the topology defined on K. In a theorem of S. Warner, necessary and sufficient conditions have been given for the neighborhood filter of zero in A to be a fundamental system of neighborhoods of zero for a topology on K. Moreover K, with this topology, is a topological field with A topologically embedded in K as an open set. For rings satisfying the conditions of this theorem, the topology for K which is defined in this paper is shown to reduce to that specified by Warner.

Mathematical Subject Classification
Primary: 16.98
Received: 15 February 1970
Published: 1 March 1971
Joby Milo Anthony