Quintasymptotic sequences over an ideal and quintasymptotic cograde (Q274560)

Let \(R\) be a Noetherian commutative ring with identity and \(I\) an ideal of \(R\). The main aim of this paper is to construct a theory of quintasymptotic sequences and quintasymptotic cograde parallel to the existing theory of quintessential sequences and quintessential cograde. A prime ideal \(\mathfrak p\) of \(R\) is called a quintasymptotic prime ideal of \(I\) if there exists a minimal associated prime \(z\) of \({\widehat{R_{\mathfrak p}}}\) such that \(\mathrm{Rad}(I\widehat{R_{\mathfrak p}}+z)={\mathfrak p}\widehat{R_{\mathfrak p}}\). The set of quitasymptotic primes of \(I\) is denoted by \(\overline{Q^*}(I)\). A sequence \(x=x_1,\dots ,x_n\) of elements of \(R\) is called a quintasymptotic sequence over \(I\) if \(I+<x_1,\dots ,x_n>\subsetneqq R\) and \[ x_i\notin \bigcup \{\mathfrak p|\mathfrak p\in \overline{Q^*}(I+<x_1,\dots ,x_{i-1}>)\} \] for all \(i=1,\dots ,n\). The authors prove that if \(R\) is local, then any two maximal quintasymptotic sequences over \(I\) have the same length. If \(\mathrm{qacogd}(I)\) denotes this common length, then \[ \mathrm{qacogd}(I)=\min\{\dim \widehat{R}/I\widehat{R}+z|z\;\text{is a minimal associated prime ideal of }\widehat{R}\}. \] They also establish many other nice equalities and inequalities for the quantity \(\mathrm{qacogd}(I)\).