This note gives a unifying characterization and exposition of strongly irreducible elements and their duals in lattices. The interest in the study of strong irreducibility stems from commutative ring theory, while the dual concept of strong irreducibility had been used to define Zariski-like topologies on specific lattices of submodules of a given module over an associative ring. Based on our lattice theoretical approach, we give a unifying treatment of strong irreducibility, dualize results on strongly irreducible submodules, examine its behavior under central localization and apply our theory to the frame of hereditary torsion theories.
Examples and Counterexamples in Commutative Ring Theory