Tidy Noether lattices (Q1095170)

A Noether lattice \({\mathcal L}\) satisfying the union condition on primes which is not a domain and in which every nonzero principal element is integrally closed is characterized in terms of its direct summands. It is shown that either: \((1)\quad if\) \({\mathcal L}\) has no proper nonzero direct summands, then every nonzero principal element of \({\mathcal L}\) is integrally closed if and only if \({\mathcal L}\) is a local Noether lattice whose maximal element is principal and has square zero; or \((2)\quad if\) \({\mathcal L}\) has a proper nonzero direct summand, then every nonzero principal element of \({\mathcal L}\) is integrally closed if and only if for each minimal direct summand A of \({\mathcal L}\), the quotient lattice [0,A] is an integrally closed domain.