An Embedding Theorem for Hyperintegral Domains

Abstract

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.