We characterize those commutative rings [Formula: see text] whose ideal lattice [Formula: see text] endowed with the annihilation operation is an ortholattice. Moreover, we provide an analogous characterization for the annihilator lattice [Formula: see text] endowed with the annihilation operation. Since the ideal lattice of [Formula: see text] is modular, [Formula: see text] is already an orthomodular lattice provided it is an ortholattice. However, there exist also commutative rings whose ideal lattices are complemented but the complementation differs from annihilation. We present an example of such a ring and develop a procedure producing infinitely many rings with this property. Finally, we provide a sufficient condition for double annihilation to be a homomorphism from [Formula: see text] onto [Formula: see text].
Bi-regular rings and the ideal lattice isomorphisms
