A class of hyperrings and hyperfields

Abstract

Hyperring is a structure generalizing that of a ring, but where the addition is not a composition, but a hypercomposition, i.e., the sumx+yof two elements,x,y, of a hyperringHis, in general, not an element but a subset ofH. When the non-zero elements of a hyperring form a multiplicative group, the hyperring is called a hyperfield, and this structure generalizes that of a field. A certain class of hyperfields (residual hyperfields of valued fields) has been used by the author [1] as an important technical tool in his theory of approximation of complete valued fields by sequences of such fields. Tne non-commutative theory of hyperrings (particularly Artinian) has been studied in depth by Stratigopoulos [2].The question arises: How common are hyperrings? We prove in this paper that a conveniently defined quotientR/Gof any ringRby any normal subgroupGof its multiplicative semigroup is always a hyperring which is a hyperfield whenRis a field. We ask: Are all hyperrings isomorphic to some subhyperring of a hyperring belonging to the class just described?