Chain conditions for quasi-Frobenius extensions (Q2763924)

A quasi-Frobenius extension \(A/B\) is a ring extension such that \(A_B\) is finitely generated projective and each of the two \((B,A)\)-bimodules \(_BA_A\) and \(_B\Hom(A_B,B_B)_A\) is a direct summand of a direct sum of finitely many copies of the other.NEWLINENEWLINENEWLINEThe main result of the paper is the following Theorem: Let \(A/B\) be a quasi Frobenius extension and \(U=U_B\) a faithful injective right \(B\)-module. If \(\Hom(A_B,B_B)\) is \(U\)-torsionless as right \(B\)-module, then the quotient ring of \(A\) with respect to \(\Hom(A_B,B_B)_A\) is a quasi-Frobenius extension of the quotient ring of \(B\) with respect to \(U_B\).NEWLINENEWLINENEWLINEIt is also shown that in a quasi-Frobenius extension \(A/B\) the following holds: If \(B\) is (right-) \(U\)-Noetherian for an injective right \(B\)-module \(U\) (i.e. the set \(\text{Ann}(R,U)\) of all right ideals \(r(X)\) of \(R\) which are annihilators of subsets \(X\) of \(U\) has ACC), then \(\Hom(A_B,B_B)_A\) is injective and \(A_A\) is \(\Hom(A_B,B_B)\)-Noetherian; a result which essentially rests on \textit{C. Faith}'s well-known lemma [Nagoya Math. J. 27, 179-191 (1966; Zbl 0154.03001)], that for an injective \(R\)-module \(U_R\) the ring \(R\) is \(U\)-Noetherian iff \(U^{(N)}\) is injective.