Ideal completions and lattices of ideal

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1973

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In Section 0 we review the definition of the ideal completion of a join-semilattice, reviewing also some other needed concepts. In Section 1 we introduce the notion of an sl-semlgroup. We show that the multiplication can be extended in a unique fashion to its ideal completion so that it becomes what we call a strong sl-semlgroup. In Section 2 we consider several special subsets of the ideal completion of an sl-semlgroup and, having Introduced the notion of integrality, we look at the relations among these subsets. Finally, in the last section of Chapter I (the latter being called, naturally. Ideal Completions) we look at the ideal completion of a commutative 1-group, drawing some additional conclusions about the connections among the subsets introduced in Section 2. In Chapter II, called Prtlfer and Bezout Domains, we apply the lattice-theoretic machinery introduced in Chapter I to study the lattice of ideals of integral domains, more exactly, the lattice of submodules of their field of quotients. By using the subsets of the lattice (of submodules of the field of quotients of an integral domain) Introduced in Section 2, we define Prtlfer, Bezout, Noetherian, Dedekind and principal ideal domains and apply some of the consequences of Chapter I to this particular situation. Sections 5 and 6 are devoted to classifying completely the lattice of Ideals of Prtlfer domains. We also show at the same time that from the sl-monold point of view, there Is no difference between Prtlfer and Bezout domains, and that there Is no difference between Dedekind and principal Ideal domains. Finally, In Section 7 we present some results about lattices of modules and give still another characterization of Prtlfer domains.

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