On Finite Factorization Rings

Let R be a commutative ring with identity. R is a finite factorization ring (FFR) if every nonzero nonunit of R has only a finite number of factorizations up to order and associates. In this article, we give a characterization of R for R[X] and R[[X]] to be an FFR.     

Some Simple Derivations of k[x,y]

It is shown that the derivation y ^r ? _x  + (xy ^s  + g)? _y , where 0 ≤ r < s are integers, is a simple derivation of k[x, y], the polynomial ring in two variables over a field k of characteristic zero.     

A Non-Existence Theorem for Almost Split Sequences

Let k be a field, Q a quiver with countably many vertices and I an ideal of kQ such that kQ/I is a spectroid. In this note, we prove that there is no almost split sequence ending at an indecomposable not finitely presented representation of the bound quiver (Q, I). We then get that an indecomposable representation M of (Q, I) is the ending term of an almost split sequence if and only if it is finitely presented and not projective. The dual results are also true.     

On Quotient Rings in Alternative Rings

We introduce a notion of left nonsingularity for alternative rings and prove that an alternative ring is left nonsingular if and only if every essential left ideal is dense, if and only if its maximal left quotient ring is von Neumann regular (a Johnson-like Theorem). Finally, we obtain a Gabriel-like Theorem for alternative rings.     

The Structures of the Cohomology Rings of Wreathed 2-Groups

Let S be a wreathed 2-group. In this article we construct a free ?S-resolution of ? and describe the ring structure of the cohomology of wreathed 2-group S.     

Lie *-Nilpotence of Group Rings

Let KG be the group ring of a group G over a field K. Let * be an involution of a group G extended linearly to the group ring KG. Suppose that G is a torsion group without 2-elements and K is a field with characteristic different from 2. We prove that KG is Lie *-nilpotent if and only if KG is Lie nilpotent.     

Rings of Formal Power Series in an Infinite Set of Indeterminates

Let α be an infinite cardinal number, Λ be an index set of cardinality > α, and {X _λ}_λ∈Λ be a set of indeterminates over an integral domain D. It is well known that there are three ways of defining the ring of formal power series in {X _λ}_λ∈Λ over D, say, D[[{X _λ}]] _i for i = 1, 2, 3. In this paper, we let D[[{X _λ}]]_α = ∪ {D[[{X _λ}_λ∈Γ]]₃ | Γ ? Λ and |Γ| ≤ α}, and we then show that D[[{X _λ}]]_α is an integral domain such that D[[{X _λ}]]₂ ? D[[{X _λ}]]_α ? D[[{X _λ}]]₃. We also prove that (1) D is a Krull domain if and only if D[[{X _λ}]]_α is a Krull domain and (2) D[[{X _λ}]]_α is a unique factorization domain (UFD) (resp., π-domain) if and only if D[[X ₁,…, X _n ]] is a UFD (resp., π-domain) for every integer n ≥ 1.     

On FC-Purity and I-Purity of Modules and Köthe Rings

In this article, several characterizations of certain classes of rings via FC-purity and I-purity are considered. Among others results, it is shown that every I-pure injective left R-module is projective if and only if every FC-pure projective left R-module is injective, if and only if, R is a semisimple ring. In particular, the structures of FC-pure projective and I-pure projective modules over a left Artinian ring are completely described. Also, it is shown that every left R-module is FC-pure projective if and only if every indecomposable left R-module is a finitely presented cyclic R-module, if and only if, R is a left Köthe ring. Finally, we introduce FC-pure flatness and I-pure flatness of modules and several characterizations of these notions are given. In particular, we show that a commutative ring R is quasi-Frobenius if and only if R is an Artinian ring and I-pure injective, if and only if, R is an Artinian ring and the injective envelope E(R) is an FC-pure projective R-module.     

Right Centralizers of Semiprime Rings

Let R be a semiprime ring with Q _ml (R) the maximal left ring of quotients of R. Suppose that T: R → Q _ml (R) is an additive map satisfying T(x ²) = xT(x) for all x ∈ R. Then T is a right centralizer; that is, there exists a ∈ Q _ml (R) such that T(x) = xa for all x ∈ R.