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A class of torsion-free abelian groups characterized by the ranks of their socles

Ulrich F. Albrecht, Anthony Giovannitti, H. Pat Goeters (2002)

Czechoslovak Mathematical Journal

Butler groups formed by factoring a completely decomposable group by a rank one group have been studied extensively. We call such groups, bracket groups. We study bracket modules over integral domains. In particular, we are interested in when any bracket R -module is R tensor a bracket group.

An elementary proof of the Briançon-Skoda theorem

Jacob Sznajdman (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

We give an elementary proof of the Briançon-Skoda theorem. The theorem gives a criterionfor when a function φ belongs to an ideal I of the ring of germs of analytic functions at 0 n ; more precisely, the ideal membership is obtained if a function associated with φ and I is locally square integrable. If I can be generated by m elements,it follows in particular that I min ( m , n ) ¯ I , where J ¯ denotes the integral closure of an ideal J .

Associated primes, integral closures and ideal topologies

Reza Naghipour (2006)

Colloquium Mathematicae

Let ⊆ be ideals of a Noetherian ring R, and let N be a non-zero finitely generated R-module. The set Q̅*(,N) of quintasymptotic primes of with respect to N was originally introduced by McAdam. Also, it has been shown by Naghipour and Schenzel that the set A * a ( , N ) : = n 1 A s s R R / ( ) a ( N ) of associated primes is finite. The purpose of this paper is to show that the topology on N defined by ( ) a ( N ) : R n 1 is finer than the topology defined by ( ) a ( N ) n 1 if and only if A * a ( , N ) is disjoint from the quintasymptotic primes of with respect to N. Moreover, we show...

Clôture intégrale des idéaux et équisingularité

Monique Lejeune-Jalabert, Bernard Teissier (2008)

Annales de la faculté des sciences de Toulouse Mathématiques

This text has two parts. The first one is the essentially unmodified text of our 1973-74 seminar on integral dependence in complex analytic geometry at the Ecole Polytechnique with J-J. Risler’s appendix on the Łojasiewicz exponents in the real-analytic framework. The second part is a short survey of more recent results directly related to the content of the seminar.The first part begins with the definition and elementary properties of the ν ¯ order function associated to an ideal I of a reduced analytic...

Fiber cones and the integral closure of ideals.

R. Hübl, C. Huneke (2001)

Collectanea Mathematica

Let (R,m) be a Noetherian local ring and let I C R be an ideal. This paper studies the question of when m I is integrally closed. Particular attention is focused on the case R is a regular local ring and I is a reduced ideal. This question arose through a question posed by Eisenbud and Mazur on the existence of evolutions.

Integral closures of ideals in the Rees ring

Y. Tiraş (1993)

Colloquium Mathematicae

The important ideas of reduction and integral closure of an ideal in a commutative Noetherian ring A (with identity) were introduced by Northcott and Rees [4]; a brief and direct approach to their theory is given in [6, (1.1)]. We begin by briefly summarizing some of the main aspects.

On algebraic closures.

R. Raphael (1992)

Publicacions Matemàtiques

This is a description of some different approaches which have been taken to the problem of generalizing the algebraic closure of a field. Work surveyed is by Enoch and Hochster (commutative algebra), Raphael (categories and rings of quotients), Borho (the polynomial approach), and Carson (logic).Later work and applications are given.

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