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On primary factorizations. (English) Zbl 0665.13004

In the first part of the paper some properties of the product of primary ideals are investigated. It is proved that if \(Q_ 1\) and \(Q_ 2\) are P-primary and \(Q_ 1\) is invertible then \(Q_ 1Q_ 2\) is P-primary. Moreover, the product \(I=Q_ 1...Q_ n\) is a reduced primary representation if \(Q_ i\) is \(P_ i\)-primary, \(P_ i\neq P_ j\) and none of \(Q_ i\) may be deleted from this product. The authors prove that for Q-domains (i.e. every ideal is a product of primary ideals) every proper ideal has a unique reduced primary representation.
In the second part the authors investigate weakly factorial domains, i.e. domains in which every nonunit is a product of primary elements. They prove some equivalent statements under which a one-dimensional Noetherian domain is weakly factorial and investigate some additional properties of weakly factorial domains (GCD domains, Krull domains, factorial domains).
Finally, it is proved that R is a weakly factorial GCD domain iff the group of divisibility G(R) of R is order isomorphic to an order direct sum of rank one totally ordered abelian groups.
Reviewer: J.Močkoř

MSC:

13A15 Ideals and multiplicative ideal theory in commutative rings
13A05 Divisibility and factorizations in commutative rings
13F15 Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial)
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References:

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