Strongly graded left FTF rings (Q1802381)

A ring \(R\) (with identity) is called a left FTF ring if the class of all submodules of flat left \(R\)-modules is closed under direct products and injective hulls. For example, regular rings, quasi-Frobenius rings and semiprime left and right Goldie rings are left (and right) FTF and left IF rings are left FTF. (A ring \(R\) is left IF if every injective left \(R\)-module is flat.) FTF rings have been studied previously by the authors [in Commun. Algebra 19, 803-827 (1991; Zbl 0726.16020)]. In the paper under review, their main result states that if \(R\) is a ring strongly graded by a locally finite group \(G\) and \(R_ e\) denotes its base ring (corresponding to the identity \(e\) of \(G\)), then \(R\) is a left FTF ring if and only if \(R_ e\) is left FTF. The proof involves studying the hereditary torsion theories associated with left FTF rings. As one of several consequences, the authors show that if \(R\), \(R_ e\) and \(G\) are as above, then \(R\) is left IF if and only if \(R_ e\) is left IF.