Let $T=\biggl(\begin{matrix} A&0\\U&B\end{matrix}\biggr)$ be a formal triangular matrix ring, where $A$ and $B$ are rings $U$ is $(B, A)$-bimodule. We first give some computing formulas of projective, injective, flat $FP$-injective dimensions special left $T$-modules. Then we establish (weak) global $T$. It proven that (1) If $U_{A}$ $_{B}U$ $lD(A)\neq lD(B)$, then $lD(T)={\rm m...

Throughout this paper, k is a field, R is an algebra over k, and H is a Hopf algebra over k. We say that R# σ H is the crossed product of R and H if R# σ H becomes an algebra over k by multiplication: (a#h)(b#g) = h,g a(h 1 · b)σ(h 2 , g 1)#h 3 g 2 Let lpd(R M), lid(R M) and lf d(R M) denote the left projective dimension, left injective dimension and left flat dimension of left R-module M, resp...

Let Λ be an Auslander’s 1-Gorenstein Artinian algebra with global dimension two. If Λ admits a trivial maximal 1-orthogonal subcategory of modΛ, then for any indecomposable module M ∈ modΛ, we have that the projective dimension of M is equal to one if and only if so is its injective dimension and that M is injective if the projective dimension of M is equal to two. In this case, we further get ...

We recall a version of the Osofsky-Smith theorem in the context of a Grothendieck category and derive several consequences of this result. For example, it is deduced that every locally finitely generated Grothendieck category with a family of completely injective finitely generated generators is semisimple. We also discuss the torsion-theoretic version of the classical Osofsky theorem which cha...

For a left and right Noetherian ring $R$, we give some equivalent characterizations for $\_RR$ satisfying the Auslander condition in terms of flat (resp. injective) dimensions minimal injective coresolution resolution) $R$-modules. Furthermore, prove that an artin algebra $R$ condition, is Gorenstein if only subcategory consisting finitely generated modules contravariantly finite. As applicatio...

Let Λ be an Auslander’s 1-Gorenstein Artinian algebra with global dimension 2. If Λ admits a trivial maximal 1-orthogonal subcategory of modΛ, then for any indecomposable module M ∈ modΛ, we have that the projective dimension of M is equal to 1 if and only if so is its injective dimension and that M is injective if the projective dimension of M is equal to 2, which implies that Λ is almost here...