On quasi-ideals and bi-ideals in ternary semirings.
We continue the study of ditalgebras, an acronym for "differential tensor algebras", and of their categories of modules. We examine extension/restriction interactions between module categories over a ditalgebra and a proper subditalgebra. As an application, we prove a result on representations of finite-dimensional tame algebras Λ over an algebraically closed field, which gives information on the extension/restriction interaction between module categories of some special algebras Λ₀, called convex...
Let be a ring. A right -module is said to be retractable if whenever is a non-zero submodule of . The goal of this article is to investigate a ring for which every right R-module is retractable. Such a ring will be called right mod-retractable. We proved that The ring is right mod-retractable if and only if each is a right mod-retractable ring for each , where is an arbitrary finite set. If is a mod-retractable ring then is a mod-retractable ring.
The following results are proved for a ring : (1) If is a fully right idempotent ring having a classical left quotient ring which is right quasi-duo, then is a strongly regular ring; (2) has a classical left quotient ring which is a finite direct sum of division rings iff is a left -ring having a reduced maximal right ideal and satisfying the maximum condition on left annihilators; (3) Let have the following properties: (a) each maximal left ideal of is either a two-sided ideal...
Right ue-rings (rings with the property of the title, i.e., with the maximality of the right socle) are investigated. It is shown that a semiprime ring R is a right ue-ring if and only if R is a regular V-ring with the socle being a maximal right ideal, and if and only if the intrinsic topology of R is non-discrete Hausdorff and dense proper right ideals are semisimple. It is proved that if R is a right self-injective right ue-ring (local right ue-ring), then R is never semiprime and is Artin semisimple...
We provide a characterization of all finite-dimensional selfinjective algebras over a field K which are socle equivalent to a prominent class of selfinjective algebras of tilted type.
We give a complete description of all finite-dimensional selfinjective algebras over an algebraically closed field whose component quiver has no short cycles.