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.

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

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 $A{*}_a(,N):={\bigcup }_{n\ge 1}As{s}_{R}R/{\left(ⁿ\right)}_a^{\left(N\right)}$ of associated primes is finite. The purpose of this paper is to show that the topology on N defined by ${{\left(ⁿ\right)}_a^{\left(N\right)}{:}_R⟨⟩}_{n\ge 1}$ is finer than the topology defined by ${{\left(ⁿ\right)}_a^{\left(N\right)}}_{n\ge 1}$ if and only if $A{*}_a(,N)$ is disjoint from the quintasymptotic primes of with respect to N. Moreover, we show...

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 $\overline{\nu }$ order function associated to an ideal $I$ of a reduced analytic...

We establish several finiteness characterizations and equations for the cardinality and the length of the set of overrings of rings with nontrivial zero divisors and integrally closed in their total ring of fractions. Similar properties are also obtained for related extensions of commutative rings that are not necessarily integral domains. Numerical characterizations are obtained for rings with some finiteness conditions afterwards.

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.

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.

Let $R$ be a commutative ring with unity. The notion of maximal non valuation domain in an integral domain is introduced and characterized. A proper subring $R$ of an integral domain $S$ is called a maximal non valuation domain in $S$ if $R$ is not a valuation subring of $S$, and for any ring $T$ such that $R\subset T\subset S$, $T$ is a valuation subring of $S$. For a local domain $S$, the equivalence of an integrally closed maximal non VD in $S$ and a maximal non local subring of $S$ is established. The relation between $dim(R,S)$ and the number...

Let $R$ be a commutative ring with unity. The notion of maximal non $\lambda$-subrings is introduced and studied. A ring $R$ is called a maximal non $\lambda$-subring of a ring $T$ if $R\subset T$ is not a $\lambda$-extension, and for any ring $S$ such that $R\subset S\subseteq T$, $S\subseteq T$ is a $\lambda$-extension. We show that a maximal non $\lambda$-subring $R$ of a field has at most two maximal ideals, and exactly two if $R$ is integrally closed in the given field. A determination of when the classical $D+M$ construction is a maximal non $\lambda$-domain is given. A necessary condition is given...

The notion of maximal non-pseudovaluation subring of an integral domain is introduced and studied. Let $R\subset S$ be an extension of domains. Then $R$ is called a maximal non-pseudovaluation subring of $S$ if $R$ is not a pseudovaluation subring of $S$, and for any ring $T$ such that $R\subset T\subset S$, $T$ is a pseudovaluation subring of $S$. We show that if $S$ is not local, then there no such $T$ exists between $R$ and $S$. We also characterize maximal non-pseudovaluation subrings of a local integral domain.

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.