The ν-Prime Ideal Sum Graph of a Nearring

Technical Summary: The $\nu$-Prime Ideal Sum Graph of a Nearring

Problem Statement
The paper addresses the need to study the interaction between algebraic and combinatorial properties of nearrings through graph-theoretic constructions. While zero-divisor graphs and ideal-based graphs have been extensively studied in ring theory, the ideal structure of nearrings is more delicate, featuring distinct notions of primeness (0-prime, 3-prime, c-prime, and equiprime). Existing literature has explored zero-divisor graphs of nearrings and graphs based on specific prime ideals, but a unified framework analyzing the sum of ideals relative to these various primeness conditions was lacking. The authors aim to define and characterize the $\nu$-prime ideal sum graph ($\nu$-PSG($N$)) for a nearring $N$, where $\nu \in \{0, 3, c, e\}$.

Methodology
The authors define the $\nu$-PSG($N$) as a graph where:

• Vertices: The set of all proper, non-trivial ideals of the nearring $N$.
• Adjacency: Two distinct vertices (ideals) $I$ and $J$ are adjacent if and only if their sum $I + J$ is a proper $\nu$-prime ideal of $N$.

The methodology involves establishing fundamental structural properties of these graphs. The authors derive necessary and sufficient conditions for connectivity and calculate or bound standard graph invariants, including the clique number ($\omega$), diameter, girth, and domination number ($\gamma$). The analysis relies on the algebraic properties of nearrings, such as the existence of maximal ideals, the behavior of ideals under homomorphisms, and specific nearring properties like the Identity for Products (IFP) and right permutability. The study includes specific examples using nearrings defined on sets like $\mathbb{Z}_2^3$ and the dihedral group $D_8$ to illustrate theoretical results.

Key Contributions and Results

1. Definition and Basic Properties:
   The paper formally introduces the $\nu$-PSG($N$) for $\nu \in \{0, 3, c, e\}$. It establishes that if a nearring is simple, the graph is undefined. If a nearring contains no non-trivial $\nu$-prime ideals, the graph is totally disconnected.

2. Connectivity and Diameter:

   • Connectivity: For unital nearrings, the paper proves that $0$-PSG($N$) is disconnected if and only if $N$ is a direct sum of two simple nearrings.
   • Diameter: For a connected $0$-PSG($N$) in a unital nearring, the diameter is bounded by 4.
   • Star Graphs: The authors characterize nearrings where the graph is a star graph: this occurs if and only if $N$ has a unique minimal ideal and every other proper, non-trivial, non-minimal ideal is maximal and $\nu$-prime.

3. Complete Graphs and Domination:

   • Completeness: Conditions are provided for $\nu$-PSG($N$) to be a complete graph, specifically requiring a unique maximal $\nu$-prime ideal $M$ such that all other proper ideals are pairwise trivially intersecting.
   • Domination: For unital nearrings, the set of all maximal ideals forms a minimal dominating set for $0$-PSG($N$), implying the domination number $\gamma(0\text{-PSG}(N)) \le |M|$.

4. Graph Invariants:

   • Girth: If $N$ contains two incomparable ideals that are adjacent, the girth of the graph is 3.
   • Clique Number: The paper establishes that the subgraph induced by a chain of $\nu$-prime ideals forms a clique. Furthermore, for a surjective nearring homomorphism $\phi: N \to N'$, the clique number satisfies $\omega(\nu\text{-PSG}(N')) \le \omega(\nu\text{-PSG}(N))$.

5. Relationships Between Variants:
   The paper clarifies the relationships between different variants of the graph based on the hierarchy of prime ideals. For instance, since every c-prime ideal is a 3-prime ideal, c-PSG($N$) is a subgraph of 3-PSG($N$). Under specific conditions (such as right permutability or IFP), these graphs can be shown to be identical.

6. Homomorphism and Isomorphism:
   The authors demonstrate that a surjective nearring homomorphism induces a graph homomorphism between the corresponding $\nu$-prime ideal sum graphs. They also provide an example of non-isomorphic nearrings that yield isomorphic $\nu$-prime ideal sum graphs, indicating that the graph structure does not uniquely determine the nearring.

Significance
The paper claims that the $\nu$-prime ideal sum graph serves as a unified framework for studying ideal-sum interactions corresponding to multiple notions of primeness in nearrings. By translating algebraic properties (such as the existence of unique maximal ideals or direct sum decompositions) into graph-theoretic properties (connectivity, diameter, and domination), the work allows for the reflection and distinction of structural features of the underlying nearring. The results provide a foundational basis for further investigation into additional graph invariants of these structures, extending the line of inquiry from ring theory into the more complex domain of nearrings.