On prime right ideals of intermediate rings of a finite normalizing extension (Q1098917)

Let R be a ring with 1. An R-bimodule M is called a finite normalizing module if there exist \(a_ 1,a_ 2,...,a_ n\) in M such that \(M=\sum^{n}_{i=1}Ra_ i=\sum^{n}_{i=1}a_ iR\). An extension ring S of R is called a finite normalizing extension if S is a finite normalizing R-bimodule. A right ideal I of R is called a prime right ideal provided that if X, Y are right ideals of R such that \(X\cdot Y\subseteq I\) then either \(X\subseteq I\) or \(Y\subseteq I\). Define \(b_ R(I)=\{x\in R:\) Rx\(\subseteq I\}.\) The main theorem of the paper is as follows: Let S be an arbitrary finite normalizing extension of R, T a ring with \(R\subset T\subset S\). If J is a prime right ideal of T, then there exist prime right ideals \(K_ 1,K_ 2,...,K_ s\) of R such that \(\cap^{s}_{i=1}K_ i=J\cap R\). In this case, \(b_ R(J\cap R)=\cap^{s}_{i=1}b_ R(K_ i)\).