Multiplicative lattices and the integral closure operation



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In 1969, J. P. Lediaev introduced the concept of the integral closure of an element in a multiplicative lattice. He showed that in the lattice of ideals of a Noetherian ring, the integral closure of an ideal coincides with its integral closure as an element of the lattice. In this dissertation, we study the integral closure operation in multiplicative lattices. In particular, attention is focused on integrally closed principal elements in multiplicative lattices, and on multiplicative lattices in which each nonzero principal element is integrally closed; these will be called tidy lattices. It is shown that integrally closed principal nonzero divisors can be characterized in terms of their prime divisors in multiplicative lattices which are Noether lattices. Then, to study tidy lattices, the concept of a direct sum of a multiplicative lattice is introduced. It is shown that tidy Noether lattices which are not domains and satisfy the union condition on primes can be characterized in terms of their minimal direct summands. Finally, the integral closure operation is used to study the prime divisors of all large powers of a fixed element of a Noether lattice.



Closure operators, Lattice theory