Gamma-Jacobson Radicals in n-ary Γ-Semirings

• Gamma-Jacobson radicals are a generalized form of the Jacobson radical for n-ary Γ-semirings, incorporating noncommutative and higher-arity structures.
• They leverage modular maximal ideals and positional conditions to introduce novel definitions of n-ary primeness and semiprimality.
• This framework enables spectral topology analyses and Wedderburn–Artin-type decompositions, advancing representation theory in complex semiring structures.

Gamma-Jacobson radicals generalize the classical Jacobson radical to the setting of noncommutative and $n$-ary $\Gamma$-semirings. The development of Gamma-Jacobson radicals is motivated by the need for a unified radical theory accommodating both higher arity and noncommutative phenomena, including positional (left/right/two-sided) structure and modular maximality constraints. These radicals allow for the characterization of semisimplicity, primitivity, and the intersection-theoretic core of $\Gamma$-semirings, both in the classical and higher-arity noncommutative cases (Gokavarapu et al., 18 Nov 2025).

1. $n$-ary $\Gamma$-Semirings and Modularity Foundations

An $n$-ary $\Gamma$-semiring is defined as a quadruple

$(T, +; \Gamma, \mu)$

where $(T, +)$ is a commutative additive semigroup with identity $0$, $\Gamma$0 is an additive semigroup, and the $\Gamma$1-ary product

$\Gamma$2

satisfies additivity and zero-absorption in each slot, along with full $\Gamma$3-ary associativity under the action of $\Gamma$4. In this setting, modular maximal ideals are those two-sided ideals admitting a quasi-unit relative to the $\Gamma$5-ary product structure—these play a pivotal role in the construction of the Gamma-Jacobson radical.

2. Ideals and (n, m)-type Structures

The foundational extensions of ideal theory to higher arity and noncommutativity lead to the concepts of left, right, and two-sided ideals. For $\Gamma$6, left ideals satisfy closure under products with the second input in the ideal, while right and two-sided ideals use analogously indexed conditions. In the general $\Gamma$7-ary case, an $\Gamma$8-ideal is a subsemigroup $\Gamma$9 such that for any nonempty $\Gamma$0, the $\Gamma$1-ary product is closed in $\Gamma$2 when all inputs indexed by $\Gamma$3 are in $\Gamma$4.

A further refinement establishes $\Gamma$5-type ideals: for $\Gamma$6, $\Gamma$7 is an $\Gamma$8-ideal if for any $\Gamma$9, the number of entries from $n$0 at least $n$1 implies closure under $n$2. The minimal such $n$3 is the arity-threshold $n$4. A decomposition theorem clarifies that every $n$5-ideal can be viewed as the intersection of all $n$6-ideals with $n$7.

3. $n$8-ary Primality, Semiprimality, and Radical Closures

Primeness and semiprimality are $n$9-ary generalizations, defined as follows:

• $\Gamma$0-ary primality: A proper $\Gamma$1-ideal $\Gamma$2 is $\Gamma$3-ary prime if $\Gamma$4 implies that some $\Gamma$5.
• $\Gamma$6-ary semiprimality: A two-sided ideal $\Gamma$7 is $\Gamma$8-ary semiprime if for all $\Gamma$9 and $n$0, $n$1 (where $n$2) implies $n$3.

The $n$4-ary prime radical of $n$5 is

$n$6

with an alternative, diagonal characterization: $n$7 This operator is a closure, and $n$8 is $n$9-ary semiprime if and only if $\Gamma$0.

4. The Gamma-Jacobson Radical: Construction and Properties

For any $\Gamma$1-ary $\Gamma$2-semiring $\Gamma$3, the Gamma-Jacobson radical is defined as

$\Gamma$4

where $\Gamma$5 is the family of all modular maximal two-sided ideals—that is, those admitting a quasi-unit.

Fundamental properties:

• $\Gamma$6 is always $\Gamma$7-ary semiprime.
• $\Gamma$8 if and only if $\Gamma$9 is $(T, +; \Gamma, \mu)$0-ary $(T, +; \Gamma, \mu)$1-semisimple.
• If every modular maximal ideal is $(T, +; \Gamma, \mu)$2-ary prime, then $(T, +; \Gamma, \mu)$3 coincides with the intersection of all maximal ideals.

A plausible implication is that this framework presents an extension of semisimplicity, primitivity, and radical theory simultaneously for commutative, noncommutative, and higher-arity semirings.

5. Zariski-Type Spectral Topologies and Triadic Spectral Geometry

The radical theory is unified via spectral topologies: for each positional direction $(T, +; \Gamma, \mu)$4 (left, right, two-sided), $(T, +; \Gamma, \mu)$5 is the set of $(T, +; \Gamma, \mu)$6-prime ideals. Closed sets are defined by

$(T, +; \Gamma, \mu)$7

with compact $(T, +; \Gamma, \mu)$8-topology, and the radical of $(T, +; \Gamma, \mu)$9 is recovered as the intersection of primes containing $(T, +)$0: $(T, +)$1 A triadic spectral diagram emerges in the fully noncommutative setting: $(T, +)$2 showing the two-sided spectrum as intermediary between left and right prime spectra.

6. Wedderburn–Artin-Type Decomposition and Representation Theory

Analogous to the classical Wedderburn–Artin theorem, if $(T, +)$3 is finite or semiprimary and $(T, +)$4, then minimal primitive (thus prime) ideals $(T, +)$5 exist with

$(T, +)$6

Each $(T, +)$7 is a primitive $(T, +)$8-semiring acting faithfully on a simple module $(T, +)$9, and the decomposition is unique up to permutation. Minimal primitive ideals are pairwise comaximal, reflecting robust representation-theoretic simplicity and allowing a tight connection between the radical and module-theoretic simplicity in noncommutative/high-arity settings.

7. Illustrative Examples and Invariants

Several example constructions clarify the landscape:

Example	Main Data (paraphrased)	Radical Consequences
Matrix semiring	$0$0, $0$1, $0$2 entrywise	Row-zero = left ideal; col-zero = right; their intersection = two-sided
Three-element system	$0$3, $0$4, $0$5, ternary product as in data	Distinct left/right prime radicals, $0$6
Pinning construction	Given central idempotent $0$7, "pin" $0$8 slots to $0$9 to reduce arity	Ideals/radicals compatible under arity-reduction
Threshold invariant	For $\Gamma$00-ary $\Gamma$01, arity-threshold $\Gamma$02 measures coordinate closure	$\Gamma$03

The examples highlight the distinctions between positional and threshold properties and demonstrate the invariance and compatibility of radical theory under arity changes and noncommutativity.

Gamma-Jacobson radicals thus unify and generalize central decomposition and primitivity results, embedding classical, noncommutative, and higher-arity radical structures within a consolidated spectral and module-theoretic framework (Gokavarapu et al., 18 Nov 2025).