The Torsion Theory Generated by M-Small Modules

Abstract

Let M be a right R-module, ${\cal M}$ the class of all M-small modules, and P a projective cover of M in $\sigma$[M]. We consider the torsion theories $\tau_{\cal M}$ = (${\cal T}_{\cal M}, {\cal F}_{\cal M}$), $\tau_V$ = (${\cal T}_V, {\cal F}_V$), and $\tau_P$ = (${\cal T}_P, {\cal F}_P$) in $\sigma$[M], where $\tau_{\cal M}$ is the torsion theory generated by ${\cal M}, \tau_V$ is the torsion theory cogenerated by ${\cal M}$, and $\tau_P$ is the dual Lambek torsion theory. We study some conditions for $\tau_{\cal M}$ to be cohereditary, stable, or split, and prove that Rej(M, ${\cal M}$) = M $\Leftrightarrow$ ${\cal F}_P$ = ${\cal M}$ (= ${\cal T}_{\cal M}$ = ${\cal F}_V$) $\Leftrightarrow$ ${\cal T}_P$ = ${\cal T}_V$ $\Leftrightarrow$ Gen _M (P) $\subseteq$ ${\cal T}_V$.