Strongly graded left FTF rings

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An associative ring R with identity is said to be a left FTF ring when the class of the submodules of flat left R-modules is closed under injective hulls and direct products .We prove (Theorem 3.5) that a strongly graded ring R by a locally finite group G is left FTF if and only if Re is left FTF, where e is the neutral element of G.This provides new examples of left FTF rings.Some consequences of this Theorem are given. . IntroductionIn the papers [5], [6] we started the study of the rings R with the property that the class of the submodules of flat left modules, .PO, is closed under direct products and injective hulls.For these rings, .Po is the torsionfree class for some hereditary torsion theory on R-Mod.Thus we can use techniques of localization relative to this torsion theory to investigate these rings.This class of rings is large and we call them left FTF rings.Clearly, QF and regular rings are left and right FTF rings .In the commutative case, Enochs showed [3] that a noetherian commutative ring R is FTF if and only if Rp is Gorestein for every minimal prime ideal P of R. Noncommutative examples of left FTF rings can be found in [5], [6] .The aim of this note is to construct new examples of left FTF rings .The tool will be the strongly graded ring theory.The paper is organized as follows .In Section 1 we record the fundamental results on left FTF rings showed in [5] and [6] which we will use in the rest of the paper.We recall also the basic machinery from torsion theories and graded rings that we will use .The main result, Theorem 2.5, appear in Section 2 .Finally, we give some applications of this result in Section 3.