The rings whose torsionfree modules have injective dimension at most one

Abstract

summary:The domains with torsion-free modules of injective dimension at most one have been examined by B. Olberding. A TF-projective module is one that is projective relative to all short exact sequences beginning with torsion-free modules. The rings, where each right ideal of $S$ is TF-projective, are precisely those rings whose torsionfree right modules have injective dimension at most one. The goal of this study is to comprehend the structure of the rings that B. Olberding recently studied. Along the way, we prove for a domain $S$, that if each ideal of $S$ is TF-projective, then $S$ is a Noetherian ring with $\dim (S)\leq 1$. Specifically, we prove that for a commutative domain $S$, each ideal of $S$ is TF-projective if and only if $S$ is a Gorenstein Dedekind domain. A left $P$-coherent ring all of its TF-projective left $S$-modules are projective is precisely left PP ring. Furthermore, we demonstrate that any (cyclic) right $S$-module of $S$ is TF-projective if and only if $S$ is a QF-ring