Some properties of TTF-classes

Abstract

Since the TTF-theory was initiated by Jams [~], various authors have made interesting studies of the theory, particularly, over semi-perfect rings. The present paper is to establish several basic properties of TTF-classes, which provide a generalization and refinement of their results. It will be shown, among others, that if (~, ~) is a torsion theory over a ring R on the left with TTF-class ~ determined by an idem~otent ideal I, then d7 is a TTF-class or hereditary if and only if I is a direct s,~d of R or ~I is flat as a right R-module respectively. Let R be a ring with unit element. A class of (unital) left R-modules is called a torsion class if it is closed under homomorphic images, direct sums, and group extensions, while it is called a torsion-free class if it is closed under submodules, direct products, and group extensions. Let C be any class of left R-modules. Define C t to be the class of those left R-modules X for which HOmR(X,M ) = 0 for all M in ~. Then ~ is a torsion class. Similarly, if we define ~r to be the class of those left R-modules Y for which HomR(M,Y ) = 0 for all M in ~, then O r is a torsion-free class. Conversely, it was shown by Dickson [3] that every torsion class and every torslon-free class is given in this way; more precisely, ~ is a torsion class if and only if (~r) 9 = ~, while ~ is a torsion-free class if and only if (d~) r = ~. A torsion class is called hereditary if it is closed under submodules. A necessary and sufficient condition for this is that the associated torsion-free class ~r is closed under inJective envelopes, as was proved also by Dickson. Now, ~ is called