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On the converse of a well-known fact about Krull domains. (English) Zbl 0694.13011

The main result in this paper gives a new characterization of a Krull domain. For any fractional ideal A of an integral domain R the author defines the v-operation and the t-operation as follows: \(A_ v=(A^{- 1})^{-1}\) where \(A^{-1}=\{x\) in the quotient field K of the domain R such that \(xA\subseteq R\}\) and \(A_ t=U\{(A)_ v\) for all finitely generated fractional ideals \(A_ 0\) contained in \(A\}.\) In case \(A=A_ v\) (or \(A_ t)\) then A is called a v-ideal (t-ideal). The author shows that if every non zero prime ideal of R contains a t-invertible prime ideal, then R is a Krull domain and gives a characterization of Krull domains in terms of t-invertibility (theorem 3.6). An integral domain is called a \(\pi\)-domain if each proper principal ideal is a finite product of prime ideals. Among several characterization of \(\pi\)-domains given (there are 4 theorems) the following is interesting namely: Every prime t-ideal is invertible.
The author concludes this nice paper with a counterexample to show that a domain R is a unique factorization domain if and only if it satisfies Krull’s principal ideal theorem.
Reviewer: N.Sankaran

MSC:

13F15 Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial)
13A05 Divisibility and factorizations in commutative rings
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
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