A NEW PROOF OF THE PERSISTENCE PROPERTY FOR IDEALS IN DEDEKIND RINGS AND PRUFER DOMAINS

In this paper, using elementary tools of commutative algebra, helps us prove the persistence property for two especial classes of rings. In fact, this paper has two main sections. In the rst 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 Ass 1 (Ik1 1 : : : Ikn n ) = Ass 1 (Ik1 1 )[


[Ass 1 (Ikn n ) for all k1; : : : ; kn  1, where for an ideal J of R, Ass 1 (J) is the stable set of associated primes of J. Moreover, we prove that every non-zero ideal in a Dedekind ring is Ratli-Rush closed, normally torsion-free and also has a strongly supercial 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 section, we prove that every non-zero nitely generated ideal in a Prufer 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.