Two applications of Nagata rings and modules

Let [Formula: see text] be a finite type hereditary torsion theory on the category of all modules over a commutative ring. The purpose of this paper is to give two applications of Nagata rings and modules in the sense of Jara [Nagata rings, Front. Math. China 10 (2015) 91–110]. First they are used to obtain Chase’s Theorem for [Formula: see text]-coherent rings. In particular, we obtain the [Formula: see text]-version of Chase’s Theorem, where [Formula: see text] is the classical star operation in ideal theory. In the second half, we apply they to characterize [Formula: see text]-flatness in the sense of Van Oystaeyen and Verschoren [Relative Invariants of Rings-The Commutative Theory, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 79 (Marcel Dekker, Inc., New York, 1983)].