FTF Rings and Frobenius Extensions

Abstract

The notion of FTF ring (see Definition 1.1 or Proposition 1.2) captures homological and finiteness properties shared by several classes of rings. Thus coherent rings with left flat-dominant dimension ≥ 1 [3, Corolario 2.2.11] or rings having quasi-Frobenius two-sided maximal quotient ring [7, Proposition 3.6; 3, Teorema 2.3.10] are examples of FTF rings. Moreover, FTF ring and QF-3 ring are related concepts (for the notion of QF-3 ring, see, e.g., [24]). For instance, any left perfect left FTF ring is QF-3 (see [3, Proposicion 2.4.1]) and, in fact, a perfect ring is FTF if and only if it is QF3 (see [7, Corollary 2.11]). More recently, FTF rings have been considered to characterize the rings for which the category of flat right R-modules is abelian [l]. Although the notion of left FTF ring is given in terms of torsion theories it can be restated without reference to the torsion theoretic framework (see Proposition 1.2). In fact, we prove in Theorem 1.8 that a ring R is a twosided FTF ring if and only if every direct product of copies of E RR and every direct product of copies of E RR is flat (here, E RR and E RR are, respectively, the injective hulls of RR and RR). In this note, we will prove that, for a Frobenius ring extension R ⊆ S in the sense of F. Kasch [14], the ring R is left FTF if and only if S is a left FTF. In fact, this result is valid (see Corollary 2.6) for more general