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.
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