Radical of splitting ring extensions.
Recently Rim and Teply [11] found a necessary condition for the existence of -torsionfree covers with respect to a given hereditary torsion theory for the category -mod. This condition uses the class of -exact modules; i.e. the -torsionfree modules for which every its -torsionfree homomorphic image is -injective. In this note we shall show that the existence of -torsionfree covers implies the existence of -exact covers, and we shall investigate some sufficient conditions for the converse....
Rim and Teply [10] investigated relatively exact modules in connection with the existence of torsionfree covers. In this note we shall study some properties of the lattice of submodules of a torsionfree module consisting of all submodules of such that is torsionfree and such that every torsionfree homomorphic image of the relative injective hull of is relatively injective. The results obtained are applied to the study of relatively exact covers of torsionfree modules. As an application...
Zelmanowitz [12] introduced the concept of ring, which we call right zip rings, with the defining properties below, which are equivalent:(ZIP 1) If the right anihilator X⊥ of a subset X of R is zero, then X1⊥ = 0 for a finite subset X1 ⊆ X.(ZIP 2) If L is a left ideal and if L⊥ = 0, then L1⊥ = 0 for a finitely generated left ideal L1 ⊆ L.In [12], Zelmanowitz noted that any ring R satisfying the d.c.c. on anihilator right ideals (= dcc ⊥) is a right zip ring, and hence, so is any subring of R. He...
We discuss the roots of the Nakayama and Auslander-Reiten translations in the derived category of coherent sheaves over a weighted projective line. As an application we derive some new results on the structure of selfinjective algebras of canonical type.
We develop the representation theory of selfinjective algebras of strictly canonical type and prove that their Auslander-Reiten quivers admit quasi-tubes maximally saturated by simple and projective modules.
We develop the representation theory of selfinjective algebras which admit Galois coverings by the repetitive algebras of algebras whose derived category of bounded complexes of finite-dimensional modules is equivalent to the derived category of coherent sheaves on a weighted projective line with virtual genus greater than one.