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Abstract

Let ${\mathfrak a}$ be a proper ideal of a commutative Noetherian ring $R$ of prime characteristic $p$ and let $Q({\mathfrak a})$ be the smallest positive integer $m$ such that $({\mathfrak a} ^{\rm F})^{[p^m]} = {\mathfrak a} ^{[p^m]}$, where ${\mathfrak a} ^{\rm F}$ is the Frobenius closure of ${\mathfrak a}$. This paper is concerned with the question whether the set $\{ Q({\mathfrak a}^{[p^m]}) : m \in {\mathbb N}_0 \}$ is bounded. We give an affirmative answer in the case that the ideal ${\mathfrak a}$ is generated by an u.s.$d$-sequence $c_1, \dots ,c_n$ for $R$ such that

(i) the map $R/\sum_{j=1}^n Rc_j\to R/\sum_{j=1}^n Rc_j^{2}$ induced by multiplication by $c_1 \dots c_n$ is an $R$-monomorphism;