International Journal of Mathematics and Mathematical Sciences
Volume 6 (1983), Issue 2, Pages 307-311

A class of hyperrings and hyperfields

Marc Krasner

Université de Paris VI, Paris 75006, France

Received 1 March 1982

Copyright © 1983 Marc Krasner. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Hyperring is a structure generalizing that of a ring, but where the addition is not a composition, but a hypercomposition, i.e., the sum x+y of two elements, x,y, of a hyperring H is, in general, not an element but a subset of H. 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 quotient R/G of any ring R by any normal subgroup G of its multiplicative semigroup is always a hyperring which is a hyperfield when R is a field. We ask: Are all hyperrings isomorphic to some subhyperring of a hyperring belonging to the class just described?