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Journal of Algebraic Systems
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Nasernejad, M. (2014). A NEW PROOF OF THE PERSISTENCE PROPERTY FOR IDEALS IN DEDEKIND RINGS AND PR¨UFER DOMAINS. Journal of Algebraic Systems, 1(2), 91-100. doi: 10.22044/jas.2014.229
Mehrdad Nasernejad. "A NEW PROOF OF THE PERSISTENCE PROPERTY FOR IDEALS IN DEDEKIND RINGS AND PR¨UFER DOMAINS". Journal of Algebraic Systems, 1, 2, 2014, 91-100. doi: 10.22044/jas.2014.229
Nasernejad, M. (2014). 'A NEW PROOF OF THE PERSISTENCE PROPERTY FOR IDEALS IN DEDEKIND RINGS AND PR¨UFER DOMAINS', Journal of Algebraic Systems, 1(2), pp. 91-100. doi: 10.22044/jas.2014.229
Nasernejad, M. A NEW PROOF OF THE PERSISTENCE PROPERTY FOR IDEALS IN DEDEKIND RINGS AND PR¨UFER DOMAINS. Journal of Algebraic Systems, 2014; 1(2): 91-100. doi: 10.22044/jas.2014.229

A NEW PROOF OF THE PERSISTENCE PROPERTY FOR IDEALS IN DEDEKIND RINGS AND PR¨UFER DOMAINS

Article 2, Volume 1, Issue 2, Winter and Spring 2014, Page 91-100  XML PDF (99 K)
Document Type: Original Manuscript
DOI: 10.22044/jas.2014.229
Author
Mehrdad Nasernejad
member of Iranian Mathematical society, Payeme Noor phd student
Abstract
In this paper, by using elementary tools of commutative algebra,
we prove the persistence property for two especial classes of rings. In fact, this
paper has two main sections. In the first main section, we let R be a Dedekind
ring and I be a proper ideal of R. We prove that if I1, . . . , In are non-zero
proper ideals of R, then Ass1(Ik1
1 . . . Ikn
n ) = Ass1(Ik1
1 ) [ · · · [ Ass1(Ikn
n )
for all k1, . . . , kn  1, where for an ideal J of R, Ass1(J) is the stable set
of associated primes of J. Moreover, we prove that every non-zero ideal in
a Dedekind ring is Ratliff-Rush closed, normally torsion-free and also has a
strongly superficial element. Especially, we show that if R = R(R, I) is the
Rees ring of R with respect to I, as a subring of R[t, u] with u = t−1, then uR
has no irrelevant prime divisor.
In the second main section, we prove that every non-zero finitely generated
ideal in a Pr¨ufer domain has the persistence property with respect to weakly
associated prime ideals. Finally, we extend the notion of persistence property
of ideals to the persistence property for rings.
Keywords
Dedekind rings; Pr¨ufer domains; weakly associated prime ideals; associated prime ideals; powers of ideals
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