## 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)

### Keywords:

pre-Schreier domain; principal ideals
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### References:

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