### $\bigcap $-compact modules

The duals of $\cup $-compact modules are briefly discussed.

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The duals of $\cup $-compact modules are briefly discussed.

It is known that a ring $R$ is left Noetherian if and only if every left $R$-module has an injective (pre)cover. We show that $\left(1\right)$ if $R$ is a right $n$-coherent ring, then every right $R$-module has an $(n,d)$-injective (pre)cover; $\left(2\right)$ if $R$ is a ring such that every $(n,0)$-injective right $R$-module is $n$-pure extending, and if every right $R$-module has an $(n,0)$-injective cover, then $R$ is right $n$-coherent. As applications of these results, we give some characterizations of $(n,d)$-rings, von Neumann regular rings and semisimple rings....

Let $\mathbb{N}$ be the set of nonnegative integers and $\mathbb{Z}$ the ring of integers. Let $\mathcal{B}$ be the ring of $N\times N$ matrices over $\mathbb{Z}$ 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 $\mathcal{B}$ yields that the subrings generated by them coincide. This subring is the sum of the ideal $\mathcal{F}$ consisting of...

∗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...

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.

We investigate the category $\text{mod}\Lambda $ of finite length modules over the ring $\Lambda =A{\otimes}_{k}\Sigma $, where $\Sigma $ is a V-ring, i.e. a ring for which every simple module is injective, $k$ a subfield of its centre and $A$ an elementary $k$-algebra. Each simple module ${E}_{j}$ gives rise to a quasiprogenerator ${P}_{j}=A\otimes {E}_{j}$. By a result of K. Fuller, ${P}_{j}$ induces a category equivalence from which we deduce that $\text{mod}\Lambda \simeq {\coprod}_{j}badhbox{P}_{j}$. As a consequence we can (1) construct for each elementary $k$-algebra $A$ over a finite field $k$ a nonartinian noetherian ring $\Lambda $ such that $\text{mod}A\simeq \text{mod}\Lambda $, (2) find twisted...

We first propose a generalization of the notion of Mathieu subspaces of associative algebras $$\mathcal{A}$$ , 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 $$\mathcal{A}$$ -modules $$\mathcal{M}$$ . 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...

Let $R$ be a left and right Noetherian ring and $C$ a semidualizing $R$-bimodule. We introduce a transpose ${\mathrm{Tr}}_{\mathrm{c}}M$ of an $R$-module $M$ with respect to $C$ 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 ${\mathrm{Tr}}_{\mathrm{c}}M$ to develop further the generalized Gorenstein dimension with respect to $C$. Especially, we generalize the Auslander-Bridger formula to the generalized...