Spectral hyperspaces of Krasner hyperrings

The purpose of this note is to prove that the hyperspaces of proper hyperideals of Krasner hyperrings are spectral.

💡 Research Summary

The paper establishes that the hyperspace of proper hyperideals of a Krasner hyperring, equipped with the lower (or “lower”) topology, is a spectral space in the sense of Hochster. After recalling the definition of a commutative unital Krasner hyperring—where addition is a multivalued hyperoperation forming a canonical hypergroup and multiplication is a commutative semigroup satisfying distributivity—the author introduces hyperideals as subsets closed under multiplication by arbitrary ring elements. The collection of all hyperideals, denoted I(R), forms a complete algebraic lattice: every hyperideal can be expressed as a sum of finitely generated (hence compact) hyperideals. This algebraic lattice structure is crucial because Hochster’s characterization of spectral spaces hinges on the underlying lattice being algebraic.

The lower topology on the set I⁺(R) of proper hyperideals is defined by taking sets of the form V_J = { I ∈ I⁺(R) | J ⊆ I } (for J ∈ I(R)) as a subbasis of closed sets. The main theorem (Theorem 1.4) asserts that (I⁺(R), τ_lower) is spectral. The proof proceeds by verifying four conditions required by Lemma 1.3, which states that a quasi‑compact, sober open subspace of a spectral space is itself spectral.

1. Spectrality of I(R). Since I(R) is an algebraic lattice, Priestley’s result (Theorem 4.2 in