Let R be a domain that is a complete local k algebra in
dimension one. In an effort to address the Berger's conjecture, a crucial
invariant reduced type s(R) was introduced by Huneke et. al. In this article,
we study this invariant and its max/min values separately and relate it to the
valuation semigroup of R. We justify the need to study s(R) in the context
of numerical semigroup rings and consequently investigate the occurrence of the
extreme values of s(R) for the Gorenstein, almost Gorenstein, and far-flung
Gorenstein complete numerical semigroup rings. Finally, we study the finiteness
of the category CM(R) of maximal Cohen Macaulay modules and the
category Ref(R) of reflexive modules for rings which are of
maximal/minimal reduced type and provide many classifications.Comment: 23 pages. Comments are welcom