Abstract
A Noether lattice 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) ifℒ has no proper nonzero direct summands, then every nonzero principal element ofℒ is integrally closed if and only if ℒ is a local Noether lattice whose maximal element is principal and has square zero; or (2) ifℒ has a proper nonzero direct summand, then every nonzero principal element ofℒ is integrally closed if and only if for each minimal direct summandA ofℒ, the quotient lattice [0,A] is an integrally closed domain.
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