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

author

Abstract:

In this paper, by using elementary tools of commutative algebra,we prove the persistence property for two especial classes of rings. In fact, thispaper has two main sections. In the first main section, we let R be a Dedekindring and I be a proper ideal of R. We prove that if I1, . . . , In are non-zeroproper ideals of R, then Ass1(Ik11 . . . Iknn ) = Ass1(Ik11 ) [ · · · [ Ass1(Iknn )for all k1, . . . , kn 1, where for an ideal J of R, Ass1(J) is the stable setof associated primes of J. Moreover, we prove that every non-zero ideal ina Dedekind ring is Ratliff-Rush closed, normally torsion-free and also has astrongly superficial element. Especially, we show that if R = R(R, I) is theRees ring of R with respect to I, as a subring of R[t, u] with u = t−1, then uRhas no irrelevant prime divisor.In the second main section, we prove that every non-zero finitely generatedideal in a Pr¨ufer domain has the persistence property with respect to weaklyassociated prime ideals. Finally, we extend the notion of persistence propertyof ideals to the persistence property for rings.

Upgrade to premium to download articles

Sign up to access the full text

Already have an account?login

similar resources

Dedekind Domains and Rings of Quotients

We study the relation of the ideal class group of a Dedekind domain A to that of As, where S is a multiplicatively closed subset of A. We construct examples of (a) a Dedekind domain with no principal prime ideal and (b) a Dedekind domain which is not the integral closure of a principal ideal domain. We also obtain some qualitative information on the number of non-principal prime ideals in an ar...

full text

a fundamental study of "histiriographic metafiction", and "literary genres", as introduced in "new historical philosophy", and tracing them in the works of julian barnes.

abstract a fundamental study of “historio-graphic metafiction” and “literary genres”, as introduced in “new historical philosophy”, and tracing them in the works of julian barnes having studied the two novels, the porcupine and arthur & george, by julian barnes, the researcher has applied linda hutcheon’s historio-graphic metafictional theories to them. the thesis is divided into five cha...

15 صفحه اول

A Property of Ideals in Polynomial Rings

Every ideal in the polynomial ring in n variables over an infinite field has a reduction generated by n elements. Eisenbud and Evans [2] proved that every ideal in k[Xx,...,Xn] can be generated up to radical by n elements (where k is a field). Avinash Sathaye [7] and Mohan Kumar [5] proved a locally complete intersection in k[ Xv ..., Xn] can be generated by n elements. In this short note we sh...

full text

A combinatorial proof of Gotzmann's persistence theorem for monomial ideals

Gotzmann proved the persistence for minimal growth for ideals. His theorem is called Gotzmann’s persistence theorem. In this paper, based on the combinatorics on binomial coefficients, a simple combinatorial proof of Gotzmann’s persistence theorem in the special case of monomial ideals is given. Introduction Let K be an arbitrary field, R = K[x1, x2, . . . , xn] the polynomial ring with deg(xi)...

full text

Math 254a: Rings of Integers and Dedekind Domains

Proof. Let (α1, . . . , αn) be any Q-basis for K; we first claim there for each i, there exists a non-zero di ∈ Z such that diαi ∈ OK . Indeed, it is easy to check that it is enough to let di be the leading term of any integer polynomial satisfied by αi. Thus, for any non-zero β ∈ I, we find that (βd1α1, . . . , βdnαn) is a Q-basis for K contained in I. It remains to show that such a basis with...

full text

Minimal Prime Ideals of Ore Extensions over Commutative Dedekind Domains

Let R = D[x;σ, δ] be an Ore extension over a commutative Dedekind domain D, where σ is an automorphism on D. In the case δ = 0 Marubayashi et. al. already investigated the class of minimal prime ideals in term of their contraction on the coefficient ring D. In this note we extend this result to a general case δ 6= 0.

full text

My Resources

Save resource for easier access later

Save to my library Already added to my library

{@ msg_add @}


Journal title

volume 1  issue 2

pages  91- 100

publication date 2014-01-01

By following a journal you will be notified via email when a new issue of this journal is published.

Hosted on Doprax cloud platform doprax.com

copyright © 2015-2023