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Let $R$ be a domain that is a complete local $\mathbb{k}$ algebra indimension one. In an effort to address the Berger's conjecture, a crucialinvariant 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 thevaluation semigroup of $R$. We justify the need to study $s(R)$ in the contextof numerical semigroup rings and consequently investigate the occurrence of theextreme values of $s(R)$ for the Gorenstein, almost Gorenstein, and far-flungGorenstein complete numerical semigroup rings. Finally, we study the finitenessof the category $\text{CM}(R)$ of maximal Cohen Macaulay modules and thecategory $\text{Ref}(R)$ of reflexive modules for rings which are ofmaximal/minimal reduced type and provide many classifications.

Extremal behavior of reduced type of one dimensional rings