Shahrood University of TechnologyJournal of Algebraic Systems2345-51281220140101A NEW PROOF OF THE PERSISTENCE PROPERTY FOR
IDEALS IN DEDEKIND RINGS AND PR¨UFER DOMAINS9110022910.22044/jas.2014.229ENMehrdad Nasernejadmember of Iranian Mathematical society, Payeme Noor phd studentJournal Article20130503In this paper, by using elementary tools of commutative algebra,<br />we prove the persistence property for two especial classes of rings. In fact, this<br />paper has two main sections. In the first main section, we let R be a Dedekind<br />ring and I be a proper ideal of R. We prove that if I1, . . . , In are non-zero<br />proper ideals of R, then Ass1(Ik1<br />1 . . . Ikn<br />n ) = Ass1(Ik1<br />1 ) [ · · · [ Ass1(Ikn<br />n )<br />for all k1, . . . , kn 1, where for an ideal J of R, Ass1(J) is the stable set<br />of associated primes of J. Moreover, we prove that every non-zero ideal in<br />a Dedekind ring is Ratliff-Rush closed, normally torsion-free and also has a<br />strongly superficial element. Especially, we show that if R = R(R, I) is the<br />Rees ring of R with respect to I, as a subring of R[t, u] with u = t−1, then uR<br />has no irrelevant prime divisor.<br />In the second main section, we prove that every non-zero finitely generated<br />ideal in a Pr¨ufer domain has the persistence property with respect to weakly<br />associated prime ideals. Finally, we extend the notion of persistence property<br />of ideals to the persistence property for rings.http://jas.shahroodut.ac.ir/article_229_aedeac2f9e82c3042ad040a8f3f9241a.pdf