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.
