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...