\(t\)-reductions and \(t\)-integral closure of ideals (Q1683393)

The authors define and study \(t\)-reductions of ideals in a commutative integral domains, in analogy to reductions of ideals introduced by Northcott and Rees. Thus, if \(I\) is a nonzero ideal of a domain \(R\), then an ideal \(J\) contained in \(I\) is a \textit{\(t\)-reduction} of \(I\), if \((JI^{n})_t=(I^{n+1})_t\) for some \(n\geq0\). Recall that \(I_t\) is the union of all ideals of the form \((R: (R: J))\), where \(J\) is a finitely generated subideal of \(I\). The authors define the \(t\)-integral closure of an ideal and prove: ``The \(t\)-integral closure of an ideal is an integrally closed ideal. In general, it is not \(t\)-closed and, a fortiori, not \(t\)-integrally closed.'' For the proof of this theorem the authors introduce the \(t\)-Rees algebra of an ideal. The authors study the persistence and contraction of integral closure under ring homomorphisms. Many examples are included in this paper.