Strong test modules and multiplier ideals (Q1407651)

Let \((R,\mathfrak{m})\) is a local ring of characteristic \(p>0\). Let \(M\) be an \(R\)-module and \(N\) a submodule in \(M\). The tight closure of \(N\) in \(M\), denoted by \(N^*_M\), is defined as follows: \(m\in N^*_M\) if there is \(c\in R-\bigcup_{\mathfrak p\in\text{Min}(R)}\mathfrak{p}\) such that \(c\otimes m\in N^{[q]}_M:= \text{Im}(F^e(N)\to F^e(M))\), for all \(e\) sufficiently large (for every \(e\), the iterated Frobenius map \(F^e:R\to R\) sends \(r\) to \(r^{p^e}\) and enables \(R\) with a new \(R\)-algebra structure on the right, denoted by \(R^e\) and \(F^e(M)\) defines as \(R^e\otimes_RM\)). Whenever \(M=R\) and \(N=I\) an ideal of \(R\), the tight closure of \(I\) in \(R\), is denoted by \(I^*\). An ideal \(J\) of \(R\) is called a strong test ideal if \(JI^*=JI\) for every ideal \(I\). The study regards the concept of strong test ideals introduced by \textit{C. Huneke} [J. Pure Appl. Algebra 122, 243--250 (1997; Zbl 0898.13003)] and have also been studied by \textit{A. Varciu} [J. Pure Appl. Algebra 167, 361-373 (2002; Zbl 0992.13002)], and \textit{N. Hara} and \textit{K.E. Smith} [Ill. J. Math. 45, 949--964 (2001; Zbl 1098.13501)]. In the paper under review the author extends the notion of strong test ideals to modules and generalizes some of the known results.