A class of hyperrings and hyperfields - Marc Krasner (International Journal of Mathematics and Mathematical Sciences)
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].