# The Torsion Theory Generated by M-Small Modules

The Torsion Theory Generated by M-Small Modules 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) $$\subseteq$$ $${\cal T}_V$$ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Algebra Colloquium Springer Journals

# The Torsion Theory Generated by M-Small Modules

, Volume 10 (1) – Jan 1, 2003
12 pages

/lp/springer-journals/the-torsion-theory-generated-by-m-small-modules-pgjnG2qcU1
Publisher
Springer Journals
Subject
Mathematics; Algebra; Algebraic Geometry
ISSN
1005-3867
eISSN
0219-1733
DOI
10.1007/s100110300006
Publisher site
See Article on Publisher Site

### 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) $$\subseteq$$ $${\cal T}_V$$ .

### Journal

Algebra ColloquiumSpringer Journals

Published: Jan 1, 2003