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

In this paper, using elementary tools of commutativealgebra, helps us prove the persistence property for two especialclasses of rings. in fact, this paper has two main sections. in therst section, we let r be a dedekind ring and i be a proper ideal ofr. we prove that if i1; : : : ; in are non-zero proper ideals of r, thenass1(ik11 : : : iknn ) = ass1(ik11 )[


[ass1(iknn ) for all k1; : : : ; kn 1, where for an ideal j of r, ass1(j) is the stable set of associatedprimes of j. moreover, we prove that every non-zero ideal in adedekind ring is ratli-rush closed, normally torsion-free and alsohas a strongly supercial element. especially, we show that ifr = r(r; i) is the rees ring of r with respect to i, as a subringof r[t; u] with u = t??1, then ur has no irrelevant prime divisor. inthe second section, we prove that every non-zero nitely generatedideal in a prufer domain has the persistence property with respectto weakly associated prime ideals. finally, we extend the notion ofpersistence property of ideals to the persistence property for rings.