Strongly Hollow Ideals: Complete Join-Primality, Dual Irreducibility, and Local Obstructions

We study strongly hollow ideals through the ideal lattice of a commutative ring. For a non-zero ideal I, let Γ(I) denote the sum of all ideals not containing I. Complete strong hollowness is shown to be exactly the intrinsic extremal condition I ⊈Γ(I), which yields principality for finitely generated strongly hollow ideals, lower-cover structure, and quotient, colon, and product permanence results. The main structural theorem gives an order-preserving duality between non-zero completely strongly hollow ideals and completely strongly irreducible ideals. We also prove local obstruction theorems governed by the Loewy layers M n /M n+1 , obtain sharp restrictions in Noetherian local rings, classify the principal ideal domain case, and establish chain-ring, distributive-lattice, and Bezout-domain criteria.