Non-Commutative n-ary Γ-Semirings

• Non-Commutative n-ary Γ-semirings are algebraic structures defined by a commutative addition and an n-ary, slot-sensitive operation modulated by a parameter semigroup Γ.
• They feature a comprehensive ideal theory, including positional, threshold, prime, and semiprime ideals with applications in spectral topology.
• The theory leverages homological and categorical frameworks, offering projective resolutions, derived functors, and a non-commutative Wedderburn–Artin decomposition.

A non-commutative $n$-ary $\Gamma$-semiring is an algebraic structure that generalizes binary $\Gamma$-semirings by encoding $n$-ary, slot-sensitive (asymmetric and non-commutative) operations modulated by a parameter semigroup $\Gamma$. This theory unifies commutative and non-commutative, binary and higher-arity frameworks, supporting a robust ideal theory, spectral topologies, and a Quillen-exact homological infrastructure that underpins non-commutative $\Gamma$-geometry. The development of these ideas has been systematized by Gokavarapu and Rao in a sequence of foundational works (Gokavarapu et al., 18 Nov 2025, Gokavarapu, 26 Nov 2025, Gokavarapu, 25 Nov 2025).

1. Formal Definition and Foundations

Let $n \geq 3$ and let $\Gamma$ be an additive semigroup (or monoid). An $n$-ary non-commutative $\Gamma$-semiring is a quadruple $(T,+,\Gamma,\mu)$ where $(T,+,0)$ is a commutative semigroup with zero, and

$\mu: T^n \times \Gamma^{n-1} \to T,$

with the $n$-ary operation denoted

$$\mu(x_1, \alpha_1, x_2, \alpha_2, \ldots, \alpha_{n-1}, x_n) = x_1{}_{\alpha_1}x_2{}_{\alpha_2}\cdots{}_{\alpha_{n-1}}x_n.$$

The structure satisfies:

• Additivity (A1): In each $T$-slot,

$$\mu(\ldots, x_i + x_i', \ldots; \vec{\alpha}) = \mu(\ldots, x_i, \ldots; \vec{\alpha}) + \mu(\ldots, x_i', \ldots; \vec{\alpha}).$$

• Zero-absorption (A2): If any $x_j = 0$, then $\mu(x_1, \ldots, x_n; \vec{\alpha}) = 0$.
• $n$-ary associativity (A3): All nestings of $\mu$ agree, i.e., all fully parenthesized words in the $\mu$-letters coincide.
• Asymmetry/Non-commutativity (A4): The operation is position-sensitive; in general, permuting the $x_i$’s changes the result:

$$x_1{}_{\alpha_1}x_2{}_{\alpha_2}\cdots{}_{\alpha_{n-1}}x_n \neq \text{any other permutation}.$$

For $\Gamma$ with zero, analogous additivity and zero-absorption in $\Gamma$-slots apply (Gokavarapu, 26 Nov 2025, Gokavarapu, 25 Nov 2025).

2. Ideal Structure: Left, Right, and $(n,m)$-Type Ideals

The non-commutative, $n$-ary context necessitates positional, threshold, and combinatorial generalizations of ideals:

• $(n,S)$-ideal (positional ideal): For $S \subseteq\{1,\dots,n\}$, $I \subseteq T$ is an $(n,S)$-ideal if $(I, +)$ is a subsemigroup and inserting elements from $I$ into slots $S$ implies that the result of $\mu$ also lies in $I$.
• Left, right, two-sided ideals: For $n=3$, $S = \{2\}$ (left), $S = \{3\}$ (right), $S = \{2,3\}$ (two-sided).
• $(n,m)$-ideals (threshold ideals): $I$ is an $(n,m)$-ideal if it is closed under addition and whenever at least $m$ of $x_1,\dots,x_n$ are in $I$, then $\mu(x_1,\dots,x_n) \in I$.

It holds that

$$\bigcap_{|S|=m} (n,S)\text{-ideals} = \text{the set of } (n, m)\text{-ideals},$$

and closure under intersection and sum extends distributively from the binary case (Gokavarapu et al., 18 Nov 2025).

3. Prime and Semiprime Ideals, Radicals

Primality is characterized diagonally:

• $n$-ary prime ideal: A proper $(n,1)$-ideal $P$ is $n$-ary prime if

$$\mu(x_1, \ldots, x_n; \vec{\alpha}) \in P \implies x_i \in P \text{ for some } i.$$

• $n$-ary semiprime ideal: Two-sided $Q$ is semiprime if

$$\mu(a, \alpha_1, a, \ldots, \alpha_{n-1}, a) \in Q \implies a \in Q$$

(i.e., “diagonal” criterion: $\Delta_n(a; \vec\alpha) \in Q$).

Quotient characterization: In $T/P$, $P$ two-sided, $P$ is $n$-ary prime iff nonzero classes $\bar{x}_i$ satisfy $\mu(\bar{x}_1,\dots,\bar{x}_n)=0$ only if some $\bar{x}_i=0$; i.e., no nonzero $n$-ary zero divisors.

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

$$\sqrt[n,\Gamma]{I} = \bigcap_{P \supseteq I,\ P\ \text{prime}} P = \{a \in T : \Delta_n(a;\vec{\alpha}) \in I \text{ for some } \vec{\alpha}\}.$$

Moreover, $Q$ is $n$-ary semiprime iff $Q = \sqrt[n,\Gamma]{Q}$ (Gokavarapu et al., 18 Nov 2025).

4. Radical Theory and Wedderburn–Artin-Type Decomposition

• Modular maximal ideal: $M\subset T$ is modular maximal if maximal among two-sided ideals and there exists $m\in T$ such that

$$a + a_\alpha m_\beta a = a,\quad\forall a,\ \alpha,\beta$$

• $\Gamma$-Jacobson radical:

$J_\Gamma(T) = \bigcap_{M\text{ modular max.}} M$

$J_\Gamma(T)$ is semiprime; $J_\Gamma(T)=0$ iff $T$ is $\Gamma$-semisimple.

For $T$ finite or semiprimary with $J_\Gamma(T)=0$, with minimal primitive ideals $P_1,\dotsc,P_s$,

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

Each $T/P_i$ is primitive, yielding a non-commutative Wedderburn–Artin decomposition. The minimal primitive ideals are pairwise comaximal, and the product decomposition is unique up to order (Gokavarapu et al., 18 Nov 2025).

5. Spectral Topology and Triadic Spectral Geometry

For $\eta\in\{L,R,2\}$ (left/right/two-sided), let $\Spec_\eta(T)$ denote the set of proper $\eta$-prime ideals, topologized by

$V_\eta(A) = \{P \in \Spec_\eta(T) : A \subseteq P\},\qquad D_\eta(A) = \Spec_\eta(T) \setminus V_\eta(A)$

This family forms the closed sets of a compact $T_0$ topology satisfying:

• $V_\eta(0) = \Spec_\eta(T)$, $V_\eta(T) = \varnothing$
• $V_\eta(A) \cap V_\eta(B) = V_\eta(A \cup B)$
• $\bigcup_i V_\eta(A_i) = V_\eta(\cap_i A_i)$
• $V_\eta(I) = V_\eta(\sqrt[\Gamma,\eta]{I})$

Primitive ideals arise as annihilators of simple $\Gamma$-modules and reside in $\Spec_2(T)$. There are continuous surjections

$\Spec_2(T) \xrightarrow{\pi_L} \Spec_L(T),\quad \Spec_2(T) \xrightarrow{\pi_R} \Spec_R(T)$

yielding a "triadic spectral geometry," mediating left, two-sided, and right prime spectra (Gokavarapu et al., 18 Nov 2025).

6. Homological and Categorical Structures

Categories of left, right, and bi-$\Gamma$-modules are constructed by tracking which slots the module element occupies. For $M$ a left module (slot $j$), the action is

$$\mu^{(j)}: T^{j-1} \times M \times T^{n-j} \times \Gamma^{n-1} \rightarrow M$$

Morphisms are additive maps commuting with positional actions. These categories are additive and admit a Quillen-exact structure with conflations as those short exact sequences respecting all slot actions (Gokavarapu, 26 Nov 2025, Gokavarapu, 25 Nov 2025).

• Projective/injective resolutions exist via free and cofree constructions, e.g. bar-type projective complexes

$$\mathbf{B}_r(M) = T^{\otimes_\Gamma r} \otimes^{(j,k)}_\Gamma M$$

with differentials using slotwise $n$-ary multiplication. Cofree injectives are given by

$I^0 = \Hom_\Gamma(T, M)$

with bimodule structure via the $n$-ary operation (Gokavarapu, 26 Nov 2025).

Derived functors $\Ext^r_\Gamma(M,N)$ and $\Tor_r^\Gamma(M,N)$ are constructed for bi-modules, respecting the Quillen-exact structure. The balance theorem guarantees independence of the choice of resolution, and the usual long exact sequences (for Ext and Tor) hold. Cup products (Yoneda composition) and Künneth-type spectral sequences are available; base-change isomorphisms exist for flat morphisms of $n$-ary $\Gamma$-semirings, paralleling classical homological algebra (Gokavarapu, 26 Nov 2025).

7. Non-Commutative $\Gamma$-Geometry and Examples

The non-commutative $\Gamma$-spectrum $\Spec_\Gamma^{nc}(T)$ is the set of prime two-sided $\Gamma$-ideals equipped with the Zariski topology and a structure sheaf assigned via $\Gamma$-localization. The abelian category of bi-$\Gamma$-modules is equivalent to the category of quasi-coherent "Gamma-sheaves" on $\Spec T$, and derived functors compute sheaf (co)homology. This framework yields a derived, non-commutative $\Gamma$-geometry, extending Grothendieck-type concepts beyond commutative settings (Gokavarapu, 26 Nov 2025, Gokavarapu, 25 Nov 2025).

Illustrative examples:

• $T = M_2(\mathbb{N}_0)$, $\Gamma = \{1\}$, with $a_1 b_1 c = a + b + c$ entrywise: sets of matrices by vanishing rows/columns are positional ideals.
• $T = \{0, a, b\}$, $a+a = b$, $b+b = b$, ternary product as specified: left/right prime ideals are $I_1 = \{0, a\},\ I_2 = \{0, b\}$, $J_\Gamma(T) = \{0\}$.
• $(n,m)$-ideals: e.g., for $n=4$, any subset $I$ with sum-closure and product closure if at least $3$ arguments lie in $I$ forms a $(4,3)$-ideal, but not a $(4,2)$-ideal (Gokavarapu et al., 18 Nov 2025).

This theory enables spectral and Morita-style analyses, an exact-categorical treatment of $\Gamma$-module categories, and positions higher-arity non-commutative semiring structures within the landscape of non-commutative algebraic geometry (Gokavarapu, 26 Nov 2025, Gokavarapu, 25 Nov 2025, Gokavarapu et al., 18 Nov 2025).