-compact modules
The duals of -compact modules are briefly discussed.
-injective covers, -coherent rings, and -rings
It is known that a ring is left Noetherian if and only if every left -module has an injective (pre)cover. We show that if is a right -coherent ring, then every right -module has an -injective (pre)cover; if is a ring such that every -injective right -module is -pure extending, and if every right -module has an -injective cover, then is right -coherent. As applications of these results, we give some characterizations of -rings, von Neumann regular rings and semisimple rings....
A canonical directly infinite ring
Let be the set of nonnegative integers and the ring of integers. Let be the ring of matrices over generated by the following two matrices: one obtained from the identity matrix by shifting the ones one position to the right and the other one position down. This ring plays an important role in the study of directly finite rings. Calculation of invertible and idempotent elements of yields that the subrings generated by them coincide. This subring is the sum of the ideal consisting of...
A Characterization Of Primal Noetherian Rings.
A characterization of representation-finite biserial algebras over a perfect field [Book]
A characterization of subgroup lattices of finite abelian groups.
A characterization of the Artinian modules.
A Characterization of Varieties of Associative Algebras of Exponent two
∗The first author was partially supported by MURST of Italy; the second author was par- tially supported by RFFI grant 99-01-00233.It was recently proved that any variety of associative algebras over a field of characteristic zero has an integral exponential growth. It is known that a variety V has polynomial growth if and only if V does not contain the Grassmann algebra and the algebra of 2 × 2 upper triangular matrices. It follows that any variety with overpolynomial growth has exponent at least...
A Class of Balanced Non-Uniserial Rings.
A combinatorial class number formula.
A commutativity-or-finiteness condition for rings.
A comparison of deformations and geometric study of varieties of associative algebras.
A construction for quasi-hereditary algebras
A Criterion for Finite Representation Type.
A criterion for quasi-heredity and the characteristic module.
A duality result for almost split sequences
Over an artinian hereditary ring R, we discuss how the existence of almost split sequences starting at the indecomposable non-injective preprojective right R-modules is related to the existence of almost split sequences ending at the indecomposable non-projective preinjective left R-modules. This answers a question raised by Simson in [27] in connection with pure semisimple rings.
A family of noetherian rings with their finite length modules under control
We investigate the category of finite length modules over the ring , where is a V-ring, i.e. a ring for which every simple module is injective, a subfield of its centre and an elementary -algebra. Each simple module gives rise to a quasiprogenerator . By a result of K. Fuller, induces a category equivalence from which we deduce that . As a consequence we can (1) construct for each elementary -algebra over a finite field a nonartinian noetherian ring such that , (2) find twisted...
A generalization of a theorem of Faith and Menal and applications.
A generalization of Mathieu subspaces to modules of associative algebras
We first propose a generalization of the notion of Mathieu subspaces of associative algebras , which was introduced recently in [Zhao W., Generalizations of the image conjecture and the Mathieu conjecture, J. Pure Appl. Algebra, 2010, 214(7), 1200–1216] and [Zhao W., Mathieu subspaces of associative algebras], to -modules . The newly introduced notion in a certain sense also generalizes the notion of submodules. Related with this new notion, we also introduce the sets σ(N) and τ(N) of stable...
A generalization of the Auslander transpose and the generalized Gorenstein dimension
Let be a left and right Noetherian ring and a semidualizing -bimodule. We introduce a transpose of an -module with respect to which unifies the Auslander transpose and Huang’s transpose, see Z. Y. Huang, On a generalization of the Auslander-Bridger transpose, Comm. Algebra 27 (1999), 5791–5812, in the two-sided Noetherian setting, and use to develop further the generalized Gorenstein dimension with respect to . Especially, we generalize the Auslander-Bridger formula to the generalized...