Regular local hyperrings and hyperdomains

Abstract

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} $.

Keywords

• hyperring
• hypermodule
• vectorial hyperspace
• dimension
• regular hyperring
• regular parameter element
• hyperdomain