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On a property of pre-Schreier domains. (English) Zbl 0634.13004

Let D be a commutative integral domain with unity. If every divisor x of a product ab of elements of D factors as \(x=rs\) where r divides a and s divides b, then D is called pre-Schreier. The author demonstrates that for finitely many elements \(a_ i\), \(1\leq i\leq m\), and \(b_ j\), \(1\leq j\leq n\), of a pre-Schreier domain there is the following identity for principal ideals: \[ (\cap_{i}(a_ i))\cdot (\cap_{j}(b_ j))=\cap_{i,j}(a_ ib_ j). \] This identity is valid in other domains, such as Prüfer domains and Bezout domains. The author studies the relationships between this identity and other conditions on a domain.
Reviewer: H.F.Kreimer

MSC:

13G05 Integral domains
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
13F15 Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial)
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