Strongly Hollow Ideals: Decomposition & Representations

• Strongly hollow ideals are submodules and lattice elements defined by a splitting condition that prevents nontrivial decompositions when combined with ideal actions.
• They underpin unique minimal PS-hollow representations in modules over Artinian rings, ensuring essential and incomparable summands.
• Applications include decomposing second modules and characterizing multiplicative lattices such as Prüfer and r-lattices through robust factorization properties.

Strongly hollow ideals and their generalizations occupy a significant role in abstract ideal theory and module theory, providing foundational insights into decompositions within commutative rings, modules, and more broadly, multiplicative lattices. The development of these concepts, including the pseudo strongly hollow (PS-hollow) submodule formalism, has enabled new analogues of primary and secondary decompositions, facilitated classification of module structures, and allowed the extension of classical ring-theoretical notions to rich lattice-theoretic contexts.

1. Definition and Lattice-Theoretic Characterization

In the context of module theory over commutative rings, a submodule $N$ of an $R$-module $M$ is called pseudo strongly hollow (PS-hollow) if for every ideal $I$ of $R$ and submodule $L$ of $M$, the implication holds: $$N \subseteq IM + L \implies N \subseteq IM \text{ or } N \subseteq L$$ (Equation (12) in (Abuhlail et al., 2018)). This axiom ensures that a PS-hollow submodule cannot be "split" nontrivially with respect to the action of ideals. The notion dualizes pseudo strongly irreducible submodules, extending the classical concept of strongly hollow submodules (which were defined earlier through similar splitting conditions).

For modules, the definition adapts: $M$ is called a PS-hollow module if

$M = IM + L \implies M = IM \text{ or } M = L$

(Equation (13) in (Abuhlail et al., 2018)). This property generalizes the decomposition principles for modules over Artinian rings and serves as a basis for constructing representations involving PS-hollow summands.

In a further abstraction, strongly hollow and completely strongly hollow ideals are studied in the context of multiplicative lattices—structures where the lattice operations (join and meet) and multiplication are compatible, and often every element is principal or decomposes into principal constituents. In these settings, strongly hollow elements preserve splitting properties analogously to the module-theoretic definitions but leverage the ambient multiplicative structure (Goswami et al., 21 Aug 2025).

2. Existence and Uniqueness Theorems for Minimal Representations

A central development in (Abuhlail et al., 2018) is the existence and uniqueness of minimal PS-hollow representations for modules over Artinian rings. If a module $M$ is PS-hollow representable (i.e., $M = N_1 + \cdots + N_n$ for PS-hollow submodules $N_i$), then there exists a minimal such representation with essential and pairwise incomparable summands. The construction proceeds as follows:

• Redundant submodules are eliminated.
• Submodules sharing the same associated hollow ideal $H$ are grouped.
• Comparable internal submodules $In(N_i)$ are replaced by the largest among them.

Uniqueness results (Theorems 2.17 and 2.18 in (Abuhlail et al., 2018)) assert that if

$M = N_1 + \cdots + N_n = K_1 + \cdots + K_m$

are minimal PS-hollow representations with respective main hollow ideals $\{H_1, \dots, H_n\}$ and $\{H'_1, \dots, H'_m\}$, then the sets coincide and corresponding internal submodules agree whenever $H_i = H'_j$. These are lattice-theoretic generalizations of uniqueness of decompositions familiar from classical theory.

In multiplicative lattices (Goswami et al., 21 Aug 2025), completely strongly hollow elements (those for which sublattice decompositions via maximal elements and localizations at residuals are robust) participate in similar unique minimal representations. The framework extends previous results on principal element lattices and demonstrates that decomposition and irreducibility phenomena can be systematically described for large classes of multiplicative lattices.

3. Structural Properties and Localization

The behavior of strongly hollow ideals and elements is closely tied to localization and residual operations in multiplicative lattices. In (Goswami et al., 21 Aug 2025), it is shown that strongly hollow elements can be characterized by their response to localizations at maximal elements and by the behavior of their residuals. Explicit descriptions are provided for Gelfand, semi-simple, Prüfer, and $r$-lattices, often reducing membership in the class of strongly hollow or completely strongly hollow elements to conditions on local summands or factorization properties.

The table below outlines characterizations given in different settings:

Lattice Type	Strongly Hollow Criterion	Complete Strong Hollowness
Prüfer lattice	Described via maximal elements and localizations	Explicit via residuals
$r$-lattice	Join/minimality properties of principal elements	Unique decomposition
Noether lattice	Explicit description for strongly hollow elements	Quasi-local via complete elements

Structural results from Anderson and Jayaram’s work provided the principal elements and local decomposition features necessary for these analyses, even though their terminology was distinct (Goswami et al., 21 Aug 2025).

4. Comparative Analysis and Generalizations

Strongly hollow ideals form a subset of PS-hollow submodules. In general, every strongly hollow submodule is PS-hollow, but the converse holds only in multiplication modules (modules where every submodule is generated by an ideal acting on the module) (Abuhlail et al., 2018). PS-hollow submodules thus offer greater flexibility, subsuming classical strongly hollow representations and adapting to contexts where multiplication conditions fail.

Comparatively, PS-hollow representations generalize primary and secondary decompositions, splitting modules into submodules with strong indivisibility under sums formed by the action of ideals. The duality with pseudo strongly irreducible submodules provides further reach, allowing one to frame decomposition results for a wider spectrum of module classes.

In the lattice context, strongly hollow elements generalize irreducibility and primality beyond classical ring-theoretic notions, emphasizing splitting properties in the setting of abstract multiplicative lattices.

5. Applications and Examples

PS-hollow and strongly hollow notions have yielded tools for decomposing modules and lattice elements in several non-classical contexts:

• Second modules: Every submodule is PS-hollow, and thus any second module is PS-hollow in itself ((Abuhlail et al., 2018), Example 2.3).
• Prüfer group $\mathbb{Z}(p^\infty)$: All submodules are strongly hollow and hence PS-hollow, though the module is not a multiplication module ((Abuhlail et al., 2018), Example 2.5(1)).
• Artinian ring case: For $R = \mathbb{Z}/pq$, $M = R[x]$, the minimal PS-hollow representation $M = pM + qM$ does not correspond to strongly hollow summands, illustrating the finer granularity afforded by PS-hollow decompositions ((Abuhlail et al., 2018), Example 2.12).

In multiplicative lattices (Goswami et al., 21 Aug 2025), representability by strongly hollow and completely strongly hollow elements provides decomposition results for structures like $C$-lattices, semi-simple lattices, and quasi-local weak $r$-lattices, with explicit constructions dependent on the maximal element framework and residual criteria.

6. Extensions and Open Questions

Several directions for future research are proposed:

• Extending PS-hollow theory to broader classes of rings (e.g., Noetherian or non-commutative rings) to prove existence and uniqueness of minimal PS-hollow representations (Abuhlail et al., 2018).
• Examining the relationships between PS-hollow submodules and classical primary/secondary decompositions, identifying instances where frameworks coincide or diverge.
• Investigating connections between pseudo distributive modules and PS-hollow submodules, including results like Proposition 2.24 that guarantee hollowness criteria in pseudo distributive (or s-lifting) settings.
• Further development of dualities between prime and hollow concepts in lattices acted upon by posets, with potential implications for topological and categorical structure theory.

A plausible implication is that tools for recognizing and constructing hollow decompositions may lead to new classification strategies in both module and lattice theory, especially as the interplay between principal generation and minimal representation is further explored.

7. Historical Influence and Conceptual Frameworks

The foundational work of Anderson and Jayaram (“Some results on abstract commutative ideal theory,” Period. Math. Hungar. 30(1), 1–26) provided lattice-theoretic generalizations for many classical concepts, including decomposition into irreducible or principal elements, criteria for locality, and frameworks for comparing primality and irreducibility (Goswami et al., 21 Aug 2025). Their principal element paradigm underpins much of the subsequent treatment of strongly hollow and completely strongly hollow elements, influencing the development of robust decomposition theory in multiplicative lattices.

The conceptual linkage between their results and modern studies, particularly the unique representation theorems and principal generation axioms, demonstrates the centrality of lattice-theoretic perspectives to progress in hollow ideal theory and its generalizations.

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The paper of strongly hollow ideals, PS-hollow submodules, and their extensions to abstract lattice frameworks continues to drive advancements in module decomposition, ideal theory, and abstract algebraic classification, providing rigorous and flexible tools for analysis in ring, module, and lattice-theoretic contexts.