Prime Lie Rings of Derivations of Commutative Rings

Let R be a 2-torsion free commutative ring with identity, and δ a nonzero derivation of R such that R is δ-prime. Then Rδ is a prime Lie ring and any nonzero ideal of Rδ contains an ideal of the form Jδ where J is a nonzero δ-ideal of R.     

On Weak Armendariz Rings

We introduce weak Armendariz rings which are a generalization of semicommutative rings and Armendariz rings, and investigate their properties. Moreover, we prove that a ring R is weak Armendariz if and only if for any n, the n-by-n upper triangular matrix ring T _n (R) is weak Armendariz. If R is semicommutative, then it is proven that the polynomial ring R[x] over R and the ring R[x]/(x ⁿ ), where (x ⁿ ) is the ideal generated by x ⁿ and n is a positive integer, are weak Armendariz.     

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.