The connes spectrums of certain finite non-abelian groups

In this article, we investigate when every simple module has a projective (pre)envelope. It is proven that (1) every simple right R-module has a projective preenvelope if and only if the left annihilator of every maximal right ideal of R is finitely generated; (2) every simple right R-module has an epic projective envelope if and only if R is a right PS ring; (3) Every simple right R-module has a monic projective preenvelope if and only if R is a right Kasch ring and the left annihilator of every maximal right ideal of R is finitely generated.     

Algebraic Characterization of Dense-Separability Among Compact Spaces

In this note, we show that a compact Hausdorff space X is dense-separable if and only if every family of ideals of C(X) with zero intersection has a countable subfamily with zero intersection. As a consequence of this characterization we observe that every compact dense-separable space whith Soc(C(X)) = 0 has a countable dense and co-dense subset.     

Associated and Attached Primes of Some Graded Modules Over Semigroup Rings

In this paper, algebras are finite dimensional over an algebraically closed field k, and modules are k-finite dimensional left modules. We prove the stable equivalence conjecture for algebras stably equivalent to algebras A with the following conditions: basic, connected and selfinjective; rad³A = 0 but rad²A ¬ 0; and the separated quiver Q³ _A of the quiver Q_A of A consisting of more than two connected components.     

Hypercentral Unit Groups and the Hyperbolicity of a Modular Group Algebra

We classify groups G such that the unit group 𝒰 ₁(? G) is hypercentral. In the second part, we classify groups G whose modular group algebra has hyperbolic unit groups 𝒰 ₁(KG).     

A Family of Graded Modules Associated to a Module

In this article we introduce a certain family of graded modules associated to a given module. These modules provide a natural extension of the notion of the associated graded ring of an ideal. We will investigate their properties. In particular, we will try to extend Ree' theorem on the associated graded ring of an ideal generated by a regular sequence to this context.     

Commuting Graphs of Matrix Algebras

The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all noncentral elements of R, and two distinct vertices x and y are adjacent if and only if xy = yx. The commuting graph of a group G, denoted by Γ(G), is similarly defined. In this article we investigate some graph-theoretic properties of Γ(M _n (F)), where F is a field and n ≥ 2. Also we study the commuting graphs of some classical groups such as GL _n (F) and SL _n (F). We show that Γ(M _n (F)) is a connected graph if and only if every field extension of F of degree n contains a proper intermediate field. We prove that apart from finitely many fields, a similar result is true for Γ(GL _n (F)) and Γ(SL _n (F)). Also we show that for two fields F and E and integers n, m ≥ 2, if Γ(M _n (F))?Γ(M _m (E)), then n = m and |F|=|E|.     

Synthesis of Azabicyclic Building Blocks for Daphniphyllum Alkaloid Intermediates Featuring N‐Trichloroacetyl Enamide 5‐endo‐trig Radical Cyclizations

A general procedure is reported for the synthesis of cis ring fused azapolycyclic compounds bearing an all‐carbon quaternary stereocenter at the ring fusion and an adequate functionalization for the assembly of new rings leading to advanced synthetic intermediates for Daphniphyllum alkaloid synthesis. The key carbon?carbon bond‐forming step in this approach is a radical cyclization of an N‐cycloalkenyl trichloroacetamide derivative involving a tetrasubstituted enamide to achieve polyfunctionalized lactams.