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Finite character representations for integral domains. (English) Zbl 0773.13004

The authors give necessary and sufficient conditions in order that an integral domain \(D\) has a finite character representation \(\bigcap_{\text{ht}P=1}\{D_ P\}\). They prove, for example, that \(D\) has such a representation if and only if every proper principal ideal in \(D\) is a \(t\)-product of primary ideals, that is if and only if for every minimal prime ideal \(P\) in \(D\) over a proper principal ideal \((x)\), we have that \((x)_ P\cap D\) is \(t\)-invertible. This result is used to characterize some classes of integral domains. Finally the authors study when \(\bigcap_{P\in t-\text{Max}(D)}\{D_ P\}\) has finite character.

MSC:

13G05 Integral domains
13E15 Commutative rings and modules of finite generation or presentation; number of generators
13F15 Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial)
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