A (p, q) connected graph is edge-odd graceful graph if there exists an injective map f: E(G) → {1, 3, ..., 2q-1} so that induced map f+: V(G) → {0, 1,2, 3, ..., (2k-1)}defined by f+(x)  f(x, y) (mod 2k), where the vertex x is incident with other vertex y and k = max {p, q} makes all the edges distinct.

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

Let G be a finite group and k be a field of characteristic p. The connection between the modular representations and the cohomology of G has been a subject of major interest over the last several years. For example, recent study of the stable category of kG-modules StMod(kG) led Rickard [27] to introduce certain idempotent modules and functors. This has led to a theory of varieties for infinite...

If N is a submodule of the R-module M , and a ∈ R, let λa : M/N → M/N be multiplication by a. We say that N is a primary submodule of M if N is proper and for every a, λa is either injective or nilpotent. Injectivity means that for all x ∈ M , we have ax ∈ N ⇒ x ∈ N . Nilpotence means that for some positive integer n, aM ⊆ N , that is, a belongs to the annihilator of M/N , denoted by ann(M/N). ...

In this article we investigate the relations between Gorenstein projective dimensions of [Formula: see text]-modules and their socles for text]-minimal Auslander–Gorenstein algebras text]. First give a description projective-injective in terms socles. Then prove that text]-module text] has dimension at most if only its socle is cogenerated by text]-module. Furthermore, show can be characterised...

We describe the structure of semi-regular Auslander-Reiten components of artin algebras without external short paths in the module category. As an application we give a complete description of self-injective artin algebras whose Auslander-Reiten quiver admits a regular acyclic component without external short paths.