On Homomorphisms of Krasner Hyperrings By a homomorphism from a Krasner hyperring (A, +, ·) into a Krasner hyperring (A', +', ·') we mean a function ƒ: A → A' satisfying ƒ(x + y) ⊆ ƒ(x)+ ƒ(y) and ƒ(x · y) = ƒ(x) ·' ƒ(y) for all ×, y ∈ A. The kernel of ƒ, ker ƒ, is defined by ker ƒ = {x ∈ A | ƒ(x) = 0'} where 0' is the zero of (A', +', ·'). In fact, ker ƒ may be empty. In this paper, some general properties of a Krasner hyperring homomorphism with nonempty kernel are given. Various examples are also provided.

Abstract. It is well-known that every integral domain D can be embedded in a field F and F is constructed so that F is (up to isomorphism) the smallest field containing D. We extend this result to hyperintegral domains and hyperfields. A commutative Krasner hyperring (A, +, ·) is said to be a hyperintegral domain if (A \ {0}, ·) is a semigroup and a hyperfield if (A \ {0}, ·) is a group. It is shown that every hyperintegral domain (D, +, ·)can be embedded in a hyperfield (F, ⊕, ⊙) and the constructed hyperfield (F, ⊕, ⊙) is (up to isomorphism) the smallest hyperfield containing D.Mathematics Subject Classification 2000: 20N20.