Prime modules. (English) Zbl 0948.13004

From the paper: R. Y. Sharp, Y. Tiraş and M. Yassi [J. Lond. Math. Soc., II. Ser. 42, No. 3, 385-392 (1990; Zbl 0733.13001)] introduced the concepts of reduction and integral closure of an ideal \(I\) of a commutative ring \(R\) relative to a Noetherian \(R\)-module \(M\). They say that \(I\) is a reduction of the ideal \(J\) of \(R\) relative to a Noetherian \(R\)-module \(M\) if \(I\subseteq J\) and there exists \(s\in \mathbb{N}\) such that \(IJ^sM= J^{s+1}M\). An element \(x\) of \(R\) is said to be integrally dependent on \(I\) if there exists \(n\in \mathbb{N}\) such that \(x^nM \subseteq (\sum_{i=1}^n x^{n-1} I^i) M\). In fact this is the case if and only if \(I\) is a reduction of \(I+Rx\) relative to \(M\). Moreover, \[ I^-= \{y\in R: y\text{ is integrally dependent on \(I\) relative to }M\} \] is an ideal of \(R\), called the integral closure of \(I\) relative to \(M\), and is the largest ideal of \(R\) which has \(I\) as a reduction relative to \(M\). In this paper, we indicate the dependence of \(I^-\) on the Noetherian \(R\)-module \(M\) by means of the extended notation \(I^{-(M)}\).


13B21 Integral dependence in commutative rings; going up, going down
13C13 Other special types of modules and ideals in commutative rings


Zbl 0733.13001