Abstract

AbstractThis article introduces the concept of a condensed domain, that is, an integral domain R for which IJ = {ij: i ∊ I, j ∊ J} for all ideals I and J of R. This concept is used to characterize Bézout domains (resp., principal ideal domains; resp., valuation domains) in suitably larger classes of integral domains. The main technical results state that a condensed domain has trivial Picard group and, if quasilocal, has depth at most 1. Special attention is paid to the Noetherian case and related examples.

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