Regular local hyperrings and hyperdomains

Abstract

<abstract><p>This paper falls in the area of hypercompositional algebra. In particular it focuses on the class of Krasner hyperrings and it studies the regular local hyperrings. These are Krasner hyperrings $ R $ with a unique maximal hyperideal $ M $ having the dimension equal to the dimension of the vectorial hyperspace $ \frac{M}{M^2} $. The aim of the paper is to show that any regular local hyperring is a hyperdomain. For proving this, we make use of the relationship existing between the dimension of the vectorial hyperspaces related to the hyperring $ R $ and to the quotient hyperring $ \overline{R} = \frac{R}{\langle a\rangle} $, where $ a $ is an element in $ M\setminus M^2 $, and of the regularity of $ \overline{R} $.</p></abstract>