IMCS/Publications/BASM/Issues/BASM n.2 (15), 1994/

Torsion theories and subcategories of divisible modules. (English)

Authors: Caşu Alexei


We study the Morita context $(R$, \ ${}_RU_S$, \ ${}_SV_R$, \ $S)$ with trase - ideals $I=(U,V)\triangleleft R$ and $J=[V,U]\triangleleft S$. It is well known that $\Cal L_I(R~-~Mod)\cong$$\Cal L_J(S~-~Mod)$, where $\Cal L_I(R~-~Mod)$ is the lattice of all torsion theories of $R-Mod$ with Gabriel topologies containing $I$. To every torsion teory $r\in \Cal L(R-Mod)$ the subcategory $\Cal D(r)\subseteq Mod-R$ of all $r$-divisible modules is associated. We prove that the mentioned isomorphism implies a bijection between subcategories $\Cal D(r)$ \ $(r\ge r_{_I})$ of divisible modules of $Mod-R$ and subcategories $\Cal D(r^\prime)$ $(r^\prime \ge r_{_J})$ of divisible modules of $Mod-S$.

Institutul de Matematică Academia de Ştiinţe a Moldovei
str. Academiei 5, MD-2028 Chişinău, Moldova