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 $0$, $\Gamma$ is an additive semigroup, and the $n$-ary product

$$\mu: T^n \times \Gamma^{n-1} \longrightarrow T, \quad (x_1, \alpha_1, x_2, \dots, \alpha_{n-1}, x_n) \mapsto x_1{}_{\!\alpha_1}x_2{}_{\!\alpha_2}\cdots {}_{\!\alpha_{n-1}}x_n$$

satisfies additivity and zero-absorption in each slot, along with full $n$-ary associativity under the action of $\Gamma$. In this setting, modular maximal ideals are those two-sided ideals admitting a quasi-unit relative to the $n$-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 $n=3$, 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 $n$-ary case, an $(n, S)$-ideal is a subsemigroup $I$ such that for any nonempty $S \subset \{1, \ldots, n\}$, the $n$-ary product is closed in $I$ when all inputs indexed by $S$ are in $I$.

A further refinement establishes $(n, m)$-type ideals: for $m \leq n$, $I$ is an $(n, m)$-ideal if for any $x_1,\ldots,x_n$, the number of entries from $I$ at least $m$ implies closure under $\mu$. The minimal such $m$ is the arity-threshold $\tau(I)$. A decomposition theorem clarifies that every $(n, m)$-ideal can be viewed as the intersection of all $(n, S)$-ideals with $|S|=m$.

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

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

• $n$-ary primality: A proper $(n,1)$-ideal $P$ is $n$-ary prime if $\mu(x_1, \alpha_1, \dots, \alpha_{n-1}, x_n)\in P$ implies that some $x_i\in P$.
• $n$-ary semiprimality: A two-sided ideal $Q$ is $n$-ary semiprime if for all $a\in T$ and $\vec\alpha\in \Gamma^{n-1}$, $\Delta_n(a;\vec\alpha) \in Q$ (where $$\Delta_n(a;\vec\alpha):=\mu(a,\alpha_1,a,\alpha_2,\dots,\alpha_{n-1},a)$$) implies $a\in Q$.

The $n$-ary prime radical of $I$ is

$\sqrt[n,\Gamma]{I} = \bigcap_{P \supseteq I,\ P\ \text{%%%%54%%%%-ary prime}} P$

with an alternative, diagonal characterization: $$\sqrt[n,\Gamma]{I} = \left\{ a\in T: \exists\,\vec{\alpha}\in\Gamma^{n-1}\ \text{with }\Delta_n(a;\vec{\alpha})\in I \right\}$$ This operator is a closure, and $Q$ is $n$-ary semiprime if and only if $Q = \sqrt[n,\Gamma]{Q}$.

4. The Gamma-Jacobson Radical: Construction and Properties

For any $n$-ary $\Gamma$-semiring $T$, the Gamma-Jacobson radical is defined as

$$J_\Gamma^{(n)}(T) = \bigcap_{M \in \mathcal{M}_n} M$$

where $\mathcal{M}_n$ is the family of all modular maximal two-sided ideals—that is, those admitting a quasi-unit.

Fundamental properties:

• $J_\Gamma^{(n)}(T)$ is always $n$-ary semiprime.
• $J_\Gamma^{(n)}(T) = 0$ if and only if $T$ is $n$-ary $\Gamma$-semisimple.
• If every modular maximal ideal is $n$-ary prime, then $J_\Gamma^{(n)}(T)$ 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 $\eta\in\{L,R,2\}$ (left, right, two-sided), $\Spec_\eta(T)$ is the set of $\eta$-prime ideals. Closed sets are defined by

$V_\eta(A) = \{P \in \Spec_\eta(T)\mid A \subseteq P\}$

with compact $T_0$-topology, and the radical of $I$ is recovered as the intersection of primes containing $I$: $\sqrt[\Gamma,\eta]{I} = \bigcap_{P\in V_\eta(I)} P$ A triadic spectral diagram emerges in the fully noncommutative setting: $\Spec_2(T) \rightarrow \Spec_L(T), \quad \Spec_2(T) \rightarrow \Spec_R(T)$ 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 $T$ is finite or semiprimary and $J_\Gamma^{(n)}(T)=0$, then minimal primitive (thus prime) ideals $\{P_1,\ldots,P_s\}$ exist with

$T \cong \prod_{i=1}^s T/P_i$

Each $T/P_i$ is a primitive $\Gamma$-semiring acting faithfully on a simple module $M_i$, 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	$T=M_2(\mathbb N_0)$, $\Gamma = \{1\}$, $a_1\,b_1\,c=a+b+c$ entrywise	Row-zero = left ideal; col-zero = right; their intersection = two-sided
Three-element system	$T = \{0, a, b\}$, $a + a = b$, $\Gamma = \{\alpha\}$, ternary product as in data	Distinct left/right prime radicals, $J_\Gamma(T)=0$
Pinning construction	Given central idempotent $e$, "pin" $n-3$ slots to $e$ to reduce arity	Ideals/radicals compatible under arity-reduction
Threshold invariant	For $n$-ary $I$, arity-threshold $\tau(I)$ measures coordinate closure	$\tau(I) \in \{1,\ldots, n\}\cup\{\infty\}$

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).