Krasner Hyperrings: Algebra & Applications

Updated 8 February 2026

Krasner hyperrings are algebraic structures that generalize classical rings by replacing single-valued addition with a multivalued hyperoperation while retaining commutative multiplication.
They play a foundational role in tropical and absolute algebraic geometry by extending classical ideal theory with concepts like prime, primary, and φ–δ–hyperideals.
Their robust structure supports categorical stability and topological frameworks such as the Zariski topology, enabling advanced research in hyperideal and valuation theory.

A Krasner hyperring is a commutative algebraic structure generalizing classical rings by replacing ordinary addition with a multivalued hyperoperation, while retaining a (usually commutative) semigroup structure for multiplication and a nonzero identity. Originating in the work of M. Krasner and further developed by Marty and others, Krasner hyperrings are foundational for current developments in tropical geometry, algebraic geometry over hyperrings, and the study of generalizations of ideal theory such as prime, primary, and expansion/reduction-based notions of hyperideals. The structure is robust under canonical topologies (Zariski) and categorical constructions, and forms the algebraic backbone for applications including adèle class hyperrings, hyperfield extensions of $\mathbb{F}_1$ , and the formalization of absolute algebraic geometry.

1. Algebraic Structure of Krasner Hyperrings

Let $R$ be a nonempty set. A commutative Krasner hyperring $(R, +, \cdot, 0)$ consists of:

A hyperaddition $+\colon R \times R \to \mathcal{P}^*(R)$ $+ : R \times R \to P^{*} (R)$ , where $\mathcal{P}^*(R)$ $P^{*} (R)$ denotes the set of all nonempty subsets of $R$ $R$ . The pair $(R, +, 0)$ $(R, +, 0)$ is a canonical hypergroup, that is:
- Associativity/Reversibility: $a + (b + c) = (a + b) + c$ for all $a,b,c \in R$ .
- Commutativity: $a + b = b + a$ .
- Zero: There is $0 \in R$ with $a + 0 = \{a\}$ for all $a$ .
- Additive Inverses: For each $a$ , there exists unique $-a$ with $0 \in a + (-a)$ .
- Reversibility: $c \in a + b \implies b \in c - a$ and $a \in c - b$ .
Multiplication $(R, \cdot)$ is a commutative semigroup with absorbing $0$: $a \cdot 0 = 0 \cdot a = 0$ .
Distributivity for all $a, b, c \in R$ : $(b + c) \cdot a = (b \cdot a) + (c \cdot a)$ .

If $(R \setminus \{0\}, \cdot)$ is a group, the structure is called a Krasner hyperfield.

Krasner hyperrings generalize ordinary commutative rings. When the hyperaddition is singleton-valued, these reduce exactly to classical rings.

2. Hyperideals and Generalized Notions

A hyperideal $N \subset R$ is a nonempty subset satisfying:

$0 \in N$ ;
$a - b \subset N$ whenever $a, b \in N$ ;
$r\cdot a \in N$ for all $r \in R$ , $a \in N$ .

The set of all hyperideals $L(R)$ forms a complete lattice under intersection and hyperideal-sum. The theory recovers classical ideals when addition is single-valued.

Variants of hyperideals include:

Prime hyperideals: If $a b \in P$ then $a \in P$ or $b \in P$ .
Primary hyperideals: $a b \in Q$ implies $a \in Q$ or $b \in \sqrt{Q}$ .
Expansion (δ) and reduction (φ) hyperideals: enable interpolation between classical, weak, and generalized notions using maps $\phi \colon L(R) \to L(R) \cup \{\emptyset\}$ and $\delta \colon L(R) \to L(R)$ with monotonicity and compatibility properties.

Recent work further generalizes these using parameterized absorption, $n$ -ary and multi-ary (m,n)-Krasner hyperrings, and S-expansions (for S a multiplicatively closed set), producing new classes such as $n$ -ary S-prime hyperideals and $φ$ -δ-primary hyperideals (Kaya et al., 2021, Anbarloei, 2022, Anbarloei, 2023, Anbarloei, 2023).

3. Expansion and Absorption Frameworks

The development of generalized hyperideals in Krasner hyperrings is built on the systematic use of expansion (δ), reduction (φ), and related absorption properties. Key cases include:

φ–δ–primary hyperideals: $N \in L^*(R)$ is φ–δ–primary if $a \circ b \in N \setminus \phi(N)$ implies $a \in N$ or $b \in \delta(N)$ . Specializations recover prime, primary, and weakly (almost) primary notions by suitable choice of φ, δ; e.g., φ_id = identity, φ₀ = zero map, δ_id = identity expansion, δ_rad = radicalization. This parametric framework enables systematic interpolation between weak, strong, and structural classes of hyperideals (Kaya et al., 2021).
Absorbing hyperideals: In (m,n)-Krasner hyperring settings, further generalizations such as (k,n)-absorbing and φ-(k,n)-absorbing primary hyperideals are introduced. These hyperideals satisfy multi-factor absorption principles parameterized by $k, n$ , and expansion functions φ, enabling fine-grained taxonomy beyond the prime/primary dichotomy (Anbarloei, 2023).
S-prime and S-primary hyperideals: For an $n$ -ary multiplicative subset $S$ , S-prime (S-primary) hyperideals are defined via absorption relative to S, reflecting a synthetic localization and factorization theory extending structure theorems from commutative algebra to hyperring theory (Anbarloei, 2022).

4. Structure Theory and Categorical Stability

The category of Krasner hyperrings and their hyperideals is robust under various algebraic constructions.

Stability under Homomorphisms and Quotients: The property of being φ–δ–primary, S-primary, or (k,n)-absorbing primary is preserved under inverse images of suitable homomorphisms, under quotients by congruence relations or normal hyperideals, and persists under localization at multiplicative subsets (Kaya et al., 2021, Anbarloei, 2022, Anbarloei, 2022).
Spectrum and Zariski Topology: The set of prime hyperideals $\operatorname{Spec}(R)$ acquires a Zariski-type topology, where closed sets $V(I) = \{ P \in \operatorname{Spec}(R) : I \subseteq P \}$ encapsulate vanishing loci analogous to classical algebraic geometry. Fundamental results show $\operatorname{Spec}(R)$ and the full hyperspace of proper hyperideals $\mathcal{I}^+_R$ are spectral spaces in the sense of Hochster: quasi-compact, sober, and with a basis of compact open sets (Ameri et al., 9 Feb 2025, Goswami, 1 Feb 2026). Closedness, irreducibility, and topological properties directly correspond to algebraic properties such as primality and dimension.
Cartesian Products and Decomposition: In direct products $R = R_1 \times R_2$ , φ–δ–primary hyperideals are classified precisely: each such hyperideal is of the form $N_1 \times R_2$ (with $N_1$ φ₁–δ₁–primary and $\delta_2(R_2) = R_2$ ) or $R_1 \times N_2$ (with $N_2$ φ₂–δ₂–primary and $\delta_1(R_1) = R_1$ ). No proper φ–δ–primary hyperideal is a mixed intersection unless one factor is trivial (Kaya et al., 2021).

5. Hyperring Extensions, Valuation Theory, and Absolute Algebraic Geometry

Quotient Hyperrings: Given a commutative ring $R$ and a subgroup $G \subset R^\times$ , the quotient $R/G$ is a Krasner hyperring, with natural extension and contraction of hyperideals. The construction recovers the Krasner hyperfield $\mathbf{K}$ as the minimal such quotient and realizes absolute algebraic geometry over $\mathbb{F}_1$ by promoting the "field with one element" to a hyperfield (Connes et al., 2010).
Valuation Theory: Valuations on Krasner hyperfields generalize classical valuation rings. A valuation $v:F \to \Gamma \cup \{\infty\}$ satisfies $v(x y) = v(x) + v(y)$ and $v(z) \ge \min\{v(x),v(y)\}$ for $z \in x+y$ , and defines valuation subhyperrings, uniquely characterized in the case of Krasner-valued hyperfields by superiorly canonical additive hypergroup structures (Linzi, 2023). Hypervaluation hyperideals formalize the connection with primary ideal theory and integral closure (Nikmehr et al., 2021).
Interconnections with Tropical and Absolute Algebraic Geometry: Krasner hyperrings underpin geometric frameworks for tropical analytification, idempotent semirings, and the study of Zariski spectra over hyperfields, including adèle class hyperrings $\mathbb{H}_K = \mathbb{A}_K/K^\times$ and their groupoid-theoretic interpretations (Connes et al., 2010).

6. Parameterized and Hierarchical Hyperideal Theory

Recent advances develop a lattice of absorption hierarchies:

φ–(k,n)-absorbing primary hyperideals interpolate between prime, primary, absorbing, and weakly absorbing structures via parameter sets $(k, n)$ and reduction maps φ. Quotient and localization characterizations enable abstract primary decomposition analogues (Anbarloei, 2023).
(s,n)-absorbing δ(0)-hyperideals generalize both $N$ - and $J$ -type radicals in a single delta expansion framework, unifying their structure theory and closure properties (Anbarloei, 2024).
Absorbing hierarchies: Chains of inclusions exist: every primary hyperideal is strongly quasi-primary (sq-primary), every sq-primary is wsq-primary, and every wsq-primary is q-primary; each admits "absorbing" variants generalizing the number-theoretic absorption phenomena (Anbarloei, 2023).

7. Spectral and Topological Foundations

Proper hyperideals of a Krasner hyperring form a spectral hyperspace—equipped with a Zariski-type topology, this space admits:

Bases of compact open sets corresponding to hyperideal containments,
Sobriety: every irreducible closed subset is the closure of a unique point,
Spectrality: quasi-compactness and closure under finite intersections.

This robust topology enables the construction of hyperringed spaces and opens the way toward hyper-schemes, gluing local hyperring structures, and developing a geometric language matched to hyperstructure algebra (Ameri et al., 9 Feb 2025, Goswami, 1 Feb 2026).

References:

(Kaya et al., 2021): " $φ$ - $δ$ -Primary Hyperideals in Krasner Hyperrings"
(Anbarloei, 2022): "Extensions of $n$ -ary prime hyperideals via an $n$ -ary multiplicative subset in a Krasner $(m,n)$ -hyperring"
(Anbarloei, 2023): " $φ$ - $(k,n)$ -absorbing and $φ$ - $(k,n)$ -absorbing primary hyperideals in a krasner $(m,n)$ -hyperring"
(Anbarloei, 2023): "wsq-primary hyperideals of a Krasner (m,n)-hyperring"
(Ameri et al., 9 Feb 2025): "Zariski topology of (Krasner) hyperrings"
(Goswami, 1 Feb 2026): "Spectral hyperspaces of Krasner hyperrings"
(Connes et al., 2010): "The hyperring of adèle classes"
(Linzi, 2023): "Notes on valuation theory for Krasner hyperfields"
(Nikmehr et al., 2021): "Integral closures, Primary Hyperideals and Hypervaluation Hyperideals of Kranser Hyperrings"
(Anbarloei, 2024): "Merging N-hyperideals and J-hyperideals in one frame"