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 be a Dedekind ring and be a proper ideal of . We prove that if are non-zero proper ideals of , then for all , where for an ideal of , is the stable set of associated primes of . 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 is the Rees ring of with respect to , as a subring of with , then has no irrelevant prime divisor. \par In the second main section, we prove that every non-zero finitely generated ideal in a Pr\"{u}fer domain has the persistence property with respect to weakly associated prime ideals.
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
MLA
Nasernejad, M. . "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
HARVARD
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
CHICAGO
M. 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
VANCOUVER
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