| Summary: | Inspired by the concept of regular local rings in classical algebra, in this article we initiate the study of the regular parameter elements in a commutative local Noetherian hyperring. These elements provide a deep connection between the dimension of the hyperring and its primary hyperideals. Then, our study focusses on the concept of regular local hyperring <i>R</i>, with maximal hyperideal <i>M</i>, having the property that the dimension of <i>R</i> is equal to the dimension of the vectorial hyperspace <inline-formula><math display="inline"><semantics><mfrac><mi>M</mi><msup><mi>M</mi><mn>2</mn></msup></mfrac></semantics></math></inline-formula> over the hyperfield <inline-formula><math display="inline"><semantics><mfrac><mi>R</mi><mi>M</mi></mfrac></semantics></math></inline-formula>. Finally, using the regular local hyperrings, we determine the dimension of the hyperrings of fractions.
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