On essential left ideals of associative rings.
On filial rings
On FI-mono-retractable modules
We introduce the notion of FI-mono-retractable modules which is a generalization of compressible modules. We investigate the properties of such modules. It is shown that the rings over which every cyclic module is FI-mono-retractable are simple Noetherian -ring with zero socle or Artinian semisimple. The last section of the paper is devoted to the endomorphism rings of FI-retractable modules.
On Finitely Generated Flat Modules II.
On Finitely Generated Flat Modules III.
On finitely generated n-SG-projective modules
We prove that finitely generated n-SG-projective modules are infinitely presented.
On flat covers in varieties
Flat covers do not exist in all varieties. We give a necessary condition for the existence of flat covers and some examples of varieties where not all algebras have flat covers.
On Flatness and Projectivity of a Ring as a Module over a Fixed Subring.
On free ring extensions of degree n.
On generalization of injectivity
Characterizations of quasi-continuous modules and continuous modules are given. A non-trivial generalization of injectivity (distinct from -injectivity) is considered.
On generalizations of injectivity.
On generalized CS-modules
An -closed submodule of a module is a submodule for which is nonsingular. A module is called a generalized CS-module (or briefly, GCS-module) if any -closed submodule of is a direct summand of . Any homomorphic image of a GCS-module is also a GCS-module. Any direct sum of a singular (uniform) module and a semi-simple module is a GCS-module. All nonsingular right -modules are projective if and only if all right -modules are GCS-modules.
On generalized extending modules.
On generalized q.f.d. modules
A right -module is called a generalized q.f.d. module if every M-singular quotient has finitely generated socle. In this note we give several characterizations to this class of modules by means of weak injectivity, tightness, and weak tightness that generalizes the results in [sanh1], Theorem 3. In particular, it is shown that a module is g.q.f.d. iff every direct sum of -singular -injective modules in is weakly injective iff every direct sum of -singular weakly tight is weakly tight iff...
On Generalized Uniserial Rings and Decompositions that Complement Direct Summands.
On Gorenstein Rings.
On hereditary artinian rings and the pure semisimplicity conjecture: rigid tilting modules and a weak conjecture
A weak form of the pure semisimplicity conjecture is introduced and characterized through properties of matrices over division rings. The step from this weak conjecture to the full pure semisimplicity conjecture would be covered by proving that there do not exist counterexamples to the conjecture in a particular class of rings, which is also studied.
On hereditary rings and the pure semisimplicity conjecture II: Sporadic potential counterexamples
It was shown in [Colloq. Math. 135 (2014), 227-262] that the pure semisimplicity conjecture (briefly, pssC) can be split into two parts: first, a weak pssC that can be seen as a purely linear algebra condition, related to an embedding of division rings and properties of matrices over those rings; the second part is the assertion that the class of left pure semisimple sporadic rings (ibid.) is empty. In the present article, we characterize the class of left pure semisimple sporadic rings having finitely...
On infinite periodic rings.
On injective -modules.