Let $T=\bigl(\begin{smallmatrix}A&0\U\&B\end{smallmatrix}\bigr)$ be a formal triangular matrix ring, where $A$ and $B$ are rings $U$ is $(B, A)$-bimodule. We prove: (1) If $U\_{A}$ ${B}U$ have finite flat dimensions, then left $T$-module $\bigl(\begin{smallmatrix}M\_1\ M\_2\end{smallmatrix}\bigr){\varphi^{M}}$ Ding projective if only $M\_1$ $M\_2/{\operatorname{im}(\varphi^{M})}$ the morphism $...

Let φ : (R, m)→ (S, n) be a local homomorphism of commutative noetherian local rings. Suppose that M is a finitely generated S-module. A generalization of Grothendieck’s non-vanishing theorem is proved for M (i.e. the Krull dimension of M over R is the greatest integer i for which the ith local cohomology module of M with respect to m, Hi m(M), is non-zero). It is also proved that the Gorenstei...

Let be a left and right noetherian ring and mod the category of finitely generated left -modules. In this article, we show the following results. 1 For a positive integer k, the condition that the subcategory of mod consisting of i-torsionfree modules coincides with the subcategory of mod consisting of i-syzygy modules for any 1 ≤ i ≤ k is left-right symmetric. 2 If is an -Gorenstein ring and N...

Let Λ be a left and right noetherian ring and modΛ the category of finitely generated left Λ-modules. In this paper we show the following results: (1) For a positive integer k, the condition that the subcategory of modΛ consisting of i-torsionfree modules coincides with the subcategory of modΛ consisting of i-syzygy modules for any 1 ≤ i ≤ k is left-right symmetric. (2) If Λ is an Auslander rin...

It is shown that any left module A over a ring R can be written as the intersection of a downward directed system of injective submodules of an injective module; equivalently, as an inverse limit of one-to-one homomorphisms of injectives. If R is left Noetherian, A can also be written as the inverse limit of a system of surjective homomorphisms of injectives. Some questions are raised. The flat...

Let $H$ be a weak Hopf algebra that is finitely generated module over its affine center. We show has finite self-injective dimension and so the Brown--Goodearl Conjecture holds in this special setting.

Let X be a complex manifold. We prove that direct images of principal value and residue currents on are smooth outside sets small in certain sense. also the sheaf such is stalkwise injective OX-module.

Let Λ be a left and right noetherian ring and mod Λ the category of finitely generated left Λ-modules. In this paper we show the following statements: (1) For a positive integer k the condition that the subcategory of mod Λ consisting of i-torsionfree modules coincides with the subcategory of mod Λ consisting of i-syzygy modules for any 1 ≤ i ≤ k is left-right symmetric. (2) If Λ is an Auslande...

All rings considered are associative with identity and all occurring modules are unital right modules. We denote by Mod R the category of all R-modules. The spectrum of a ring R is known to be the “set” of isomorphism classes of indecomposable injective right R-modules. When R is right-Noetherian the spectrum describes all injective Rmodules, since each injective module is a direct sum of indec...

Recently, we obtained a new characterization for an orthogonal system to be simple-minded in the stable module category of any representation-finite self-injective algebra. In this paper, apply result give explicit construction s