A NEW PROOF OF THE PERSISTENCE PROPERTY FOR IDEALS IN DEDEKIND RINGS AND PR¨UFER DOMAINS

Nasernejad, M.

doi:10.22044/jas.2014.229

A NEW PROOF OF THE PERSISTENCE PROPERTY FOR IDEALS IN DEDEKIND RINGS AND PR¨UFER DOMAINS

Document Type : Original Manuscript

Author

M. Nasernejad

Department of Mathematics, Payame Noor University, P.O.Box 19395-3697, Tehran, Iran.

10.22044/jas.2014.229

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 $I_1,\ldots,I_n$ are non-zero proper ideals of $R$, then ${Ass}^{\infty}(I_1^{k_1}\ldots I_n^{k_n})={Ass}^{\infty}(I_1^{k_1})\cup\cdots\cup {Ass}^{\infty}(I_n^{k_n})$ for all $k_1,\ldots,k_n \geq 1$, where for an ideal $J$ of $R$, ${Ass}^{\infty}(J)$ 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 $\mathcal{R}=\mathcal{R}(R, I)$ is the Rees ring of $R$ with respect to $I$, as a subring of $R[t,u]$ with $u=t^{-1}$, then $u\mathcal{R}$ 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.

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