Coreflexive Modules and Semidualizing Modules with Finite Projective Dimension

Abstract

Let $R$ and $S$ be rings and $_S\omega_R$ a semidualizing bimodule. For a subclass $\mathcal{T}$ of the class of $\omega$-coreflexive modules and $n \geq 1$, we introduce and study modules of $\omega$-$\mathcal{T}$-class $n$. By using the properties of such modules, we get some equivalent characterizations for $\omega_S$ having finite projective dimension. In particular, we prove that the projective dimension of $\omega_S$ is at most $n$ if and only if any module of $\omega$-$\mathcal{T}$-class $n$ is $\omega$-coreflexive. Moreover, we get some equivalent characterizations for $\omega_S$ having finite projective dimension at most two or one in terms of the properties of (adjoint) $\omega$-coreflexive and $\omega$-cotorsionless modules. Finally, we give some partial answers to the Wakamatsu tilting conjecture.