accr pairs (Q1207532)

All rings are commutative with identity. An \(R\)-module \(M\) is said to satisfy accr if, for every submodule \(N\) of \(M\) and every finitely generated ideal \(I\) of \(R\), the ascending chain \[ {(N:I)\subseteq(N:I^ 2)\subseteq(N:I^ 3)\subseteq\dots} \] terminates. Rings and modules satisfying this condition have been studied by \textit{C.-P. Lu} [cf. Pac. J. Math. 131, No. 2, 303-318 (1988; Zbl 0664.13005) and Proc. Am. Math. Soc. 117, No. 1, 5-10 (1993; see the preceding review)]. For rings \(R\subseteq T\) (with the same identity), \((R,T)\) is called an accr pair if, for every ring \(A\) with \(R\subseteq A\subseteq T\), \(A\) satisfies accr. On the other hand, \((R,T)\) is called a Laskerian pair if, for every ring \(A\) with \(R\subseteq A\subseteq T\), every proper ideal of \(A\) is a finite intersection of primary ideals. Any Laskarian pair is an accr pair, by a theorem of Lu (op. cit.). The author examines the converse, and more generally when \((R,T)\) is an accr pair, in the following cases: (i) \(T=R[x]\); (ii) \(T\) is a field; (iii) \(R=F[x_ 1,\dots,x_ n]\), \(T=K[x_ 1,\dots,x_ n]\), where \(F\subseteq K\) are fields (and the corresponding question for formal power series); (iv) \(T=S^{-1}R\); (v) \(R\) is a field and \(T\) an affine ring over \(R\).