Vol. 65, No. 1, 1976

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Chained rings

Paul Froeschl

Vol. 65 (1976), No. 1, 47–53

All rings considered are commutative with identity. A chained ring is any ring whose set of ideals is totally ordered by inclusion. The main object of this paper is stating conditions in which every valuation overring of a given ring is a chained ring. It is shown that every valuation overring of a ring R is a chained ring if and only if the ideal of zero divisors of T(R), the total quotient ring of R, is the conductor of R, the integral closure of R, in T(R). An example is provided of a valuation ring which is not a chained ring even though its total quotient ring is a chained ring.

Mathematical Subject Classification 2000
Primary: 13A15
Received: 2 May 1975
Revised: 27 January 1976
Published: 1 July 1976
Paul Froeschl