On left QF− 3 rings

Tachikawa, Hiroyuki

doi:10.2140/pjm.1970.32.255

Cited by 28 publications

(8 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Moreover, (1) clearly implies (3). Now, (1) can be deduced from (2) and ( (1) and (5) in Theorem 1.8 can be seen as an analogue to the fact that a perfect ring R is QF-3 if and only if E R R is projective (see [23]). In fact, this can be obtained as a corollary of the theorem taking into account that the notions of FTF and QF-3 ring coincide over perfect rings [7,Corollary 2.11].…”

Section: R Is Almost Coherent and E R R Is Flatmentioning

confidence: 99%

FTF Rings and Frobenius Extensions

Gómez-Torrecillas

Torrecillas

2002

Journal of Algebra

View full text Add to dashboard Cite

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, Proposición 2.4.1]) and, in fact, a perfect ring is FTF if and only if it is QF-3 (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 R R and every direct product of copies of E R R is flat (here, E R R and E R R are, respectively, the injective hulls of R R and R R ).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

show abstract

Section: R Is Almost Coherent and E R R Is Flatmentioning

confidence: 99%

FTF Rings and Frobenius Extensions

Gómez-Torrecillas

Torrecillas

2002

Journal of Algebra

View full text Add to dashboard Cite

show abstract

“…R n ) the left (resp. right) iϋ-module R. Using the terminology of [5], we have the following definitions:…”

Section: On Dominant and Codominant Dimension Of Qf -3 Rings David A mentioning

confidence: 99%

On dominant and codominant dimension of QF− 3 rings

Hill¹

1973

Pacific J. Math.

View full text Add to dashboard Cite

“…Moreover, in this case R is a QF-Z ring (cf. [6] and [8]). For semi-primary or perfect rings; however, the situation is somewhat different.…”

mentioning

confidence: 98%

“…In this case E( R R) need not be protective and R need not be right QF-Z (cf. [3] and [8]). However, a perfect ring is QF-Z if and only if both E( R R) and E(R R ) are protective (see [8]).…”

mentioning

confidence: 98%