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Displaying similar documents to “On commutative rings whose prime ideals are direct sums of cyclics”

Pseudo-valuation rings. II

David F. Anderson, Ayman Badawi, David E. Dobbs (2000)

Bollettino dell'Unione Matematica Italiana

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Viene data una condizione sufficiente affinchè un sopra-anello di un anello di pseudo-valutazione (PVR) sia ancora un PVR. Da ciò segue che se R , M è un PVR, allora ogni sopra-anello di R è un PVR se (e soltanto se) R ⁢ u è quasi-locale per ciascun elemento u di M : M . Vari risultati sono dimostrati per un ideale primo di un anello commutativo arbitrario R , avente Z ⁢ R come insieme di zero-divisori. Per esempio, se P è un primo «forte» di R e contiene un elemento non-zero divisore di R , allora P : P è...

Conditions under which R ( x ) and R 〈 x 〉 are almost Q-rings

Hani A. Khashan, H. Al-Ezeh (2007)

Archivum Mathematicum

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All rings considered in this paper are assumed to be commutative with identities. A ring R is a Q -ring if every ideal of R is a finite product of primary ideals. An almost Q -ring is a ring whose localization at every prime ideal is a Q -ring. In this paper, we first prove that the statements, R is an almost Z P I -ring and R [ x ] is an almost Q -ring are equivalent for any ring R . Then we prove that under the condition that every prime ideal of R ( x ) is an extension of a prime ideal of R , the ring R ...

Fixed-place ideals in commutative rings

Ali Rezaei Aliabad, Mehdi Badie (2013)

Commentationes Mathematicae Universitatis Carolinae

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Let I be a semi-prime ideal. Then P ∘ ∈ Min ( I ) is called irredundant with respect to I if I ≠ ⋂ P ∘ ≠ P ∈ Min ( I ) P . If I is the intersection of all irredundant ideals with respect to I , it is called a fixed-place ideal. If there are no irredundant ideals with respect to I , it is called an anti fixed-place ideal. We show that each semi-prime ideal has a unique representation as an intersection of a fixed-place ideal and an anti fixed-place ideal. We say the point p ∈ β X is a fixed-place point if O p ( X ) is a fixed-place ideal. In...