Abstract
Let R be a left and right Noetherian ring. We introduce the notion of the torsionfree dimension of finitely generated R-modules. For any n 0, we prove that R is a Gorenstein ring with self-injective dimension at most n if and only if every finitely generated left R-module and every finitely generated right R-module have torsionfree dimension at most n, if and only if every finitely generated left (or right) R-module has Gorenstein dimension at most n. For any n 1, we study the properties of the finitely generated R-modules M with Ext i R (M, R) 0 for any 1 i n. Then we investigate the relation between these properties and the s elf-injective dimension of R.
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