Classification of split torsion torsionfree triples in module categories.
Let \(A\) be a ring with identity. A torsion torsionfree triple (a TTF-triple for short) in \(\text{Mod\,}A\) is a triple \((C,T,F)\) of three full subcategories in \(\text{Mod\,}A\) such that both \((C,T)\) and \((T,F)\) are torsion theories in \(\text{Mod\,}A\). A TTF-triple in \(\text{Mod\,}A\) is called left-split (resp. right-split) if the torion theory \((C,T)\) (resp. \((T,F)\)) splits. It is called centrally split if it is both left and right split. In this paper, in Section 2, the authors give the definitions and known results concerning TTF-triples in a module category. In Section 3, they give the classification of left-split TTF-triples in \(\text{Mod\,}A\). They present in Section 4 a partial classification of right-split TTF-triples. In Section 5, they classify the right-split TTF-triples \((C,T,F)\) in \(\text{Mod\,}A\) such that \(A_A\in F\), from which the classification of right-split TTF-triples follows.
scientific article