Partial Skew Polynomial Rings and Jacobson Rings

In this article we consider rings R with a partial action α of an infinite cyclic group G on R. We generalize the well-known results about Jacobson rings and strongly Jacobson rings in skew polynomial rings and skew Laurent polynomial rings to partial skew polynomial rings and partial skew Laurent polynomial rings.     

Baer and Quasi-Baer Properties of Skew Generalized Power Series Rings

Let R be a ring, (S, ≤) a strictly ordered monoid and ω: S → End(R) a monoid homomorphism. The skew generalized power series ring R[[S, ω]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal'cev–Neumann Laurent series rings. In this article, we study relations between the (quasi-) Baer, principally quasi-Baer and principally projective properties of a ring R, and its skew generalized power series extension R[[S, ω]]. As particular cases of our general results, we obtain new theorems on (skew) group rings, Mal'cev–Neumann Laurent series rings, and the ring of generalized power series.     

Algebraically closed noncommutative polynomial rings

Let R be a noncommutative polynomial ring over the division ring K where K has center F. Then R = K[x,σ,D]where σ is a monomorphism of K and D is a σ-derivaton K. R is called dimension finite if (K: F_σ)<∞ and (K: F_D)<∞ where F_σ is the subfield of F fixed under σand F_D is the subfied of F of D-constants. R is algebraically closed if every nonconstant polynomial in Rfactors completely into linear factors. The algebraically closed dimension finite polynomial rings are determined. s done by reducing the problem to two classes: skew polynomial rings and differential polynomial rings. Examples algebraically closed polynomial rings which are not dimensfinite are given.     

Radicals in skew polynomial and skew Laurent polynomial rings

We provide a general procedure for characterizing radical-like functions of skew polynomial and skew Laurent polynomial rings under grading hypotheses. In particular, we are able to completely characterize the Wedderburn and Levitzki radicals of skew polynomial and skew Laurent polynomial rings in terms of ideals in the coefficient ring. We also introduce the T-nilpotent radideals, and perform similar characterizations.     

Semiprimeness,quasi-Baerness and prime radical of skew generalized power series rings

Let R be a ring, (S,≤) a strictly ordered monoid and ω:S→End(R) a monoid homomorphism. The skew generalized power series ring R[[S,ω]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal’cev-Neumann Laurent series rings. In this paper we obtain necessary and sufficient conditions for the skew generalized power series ring R[[S,ω]] to be a semiprime, prime, quasi-Baer, or Baer ring. Furthermore, we study the prime radical of a skew generalized power series ring R[[S,ω]]. Our results extend and unify many existing results. In particular, we obtain new theorems on (skew) group rings, Mal’cev-Neumann Laurent series rings and the ring of generalized power series.     

The Zero-Divisor Graphs Which Are Uniquely Determined By Neighborhoods

A nonempty simple connected graph G is called a uniquely determined graph, if distinct vertices of G have distinct neighborhoods. We prove that if R is a commutative ring, then Γ(R) is uniquely determined if and only if either R is a Boolean ring or T(R) is a local ring with x² = 0 for any x ∈ Z(R), where T(R) is the total quotient ring of R. We determine all the corresponding rings with characteristic p for any finite complete graph, and in particular, give all the corresponding rings of K_n if n + 1 = p^q for some primes p, q. Finally, we show that a graph G with more than two vertices has a unique corresponding zero-divisor semigroup if G is a zero-divisor graph of some Boolean ring.     

Free Pairs of Bicyclic Units in the Integral Group Ring of the Dihedral Group

Hirano studied the quasi-Armendariz property of rings, and then this concept was generalized by some authors, defining quasi-Armendariz property for skew polynomial rings and monoid rings. In this article, we consider unified approach to the quasi-Armendariz property of skew power series rings and skew polynomial rings by considering the quasi-Armendariz condition in mixed extension ring [R; I][x; σ], introducing the concept of so-called (σ, I)-quasi Armendariz ring, where R is an associative ring equipped with an endomorphism σ and I is an σ-stable ideal of R. We study the ring-theoretical properties of (σ, I)-quasi Armendariz rings, and we obtain various necessary or sufficient conditions for a ring to be (σ, I)-quasi Armendariz. Constructing various examples, we classify how the (σ, I)-quasi Armendariz property behaves under various ring extensions. Furthermore, we show that a number of interesting properties of an (σ, I)-quasi Armendariz ring R such as reflexive and quasi-Baer property transfer to its mixed extension ring and vice versa. In this way, we extend the well-known results about quasi-Armendariz property in ordinary polynomial rings and skew polynomial rings for this class of mixed extensions. We pay also a particular attention to quasi-Gaussian rings.     

Classification of multivariate skew polynomial rings over finite fields via affine transformations of variables

In this work, free multivariate skew polynomial rings are considered, together with their quotients over ideals of skew polynomials that vanish at every point (which includes minimal multivariate skew polynomial rings). We provide a full classification of such multivariate skew polynomial rings (free or not) over finite fields. To that end, we first show that all ring morphisms from the field to the ring of square matrices are diagonalizable, and that the corresponding derivations are all inner derivations. Secondly, we show that all such multivariate skew polynomial rings over finite fields are isomorphic as algebras to a multivariate skew polynomial ring whose ring morphism from the field to the ring of square matrices is diagonal, and whose derivation is the zero derivation. Furthermore, we prove that two such representations only differ in a permutation on the field automorphisms appearing in the corresponding diagonal. The algebra isomorphisms are given by affine transformations of variables and preserve evaluations and degrees. In addition, ours proofs show that the simplified form of multivariate skew polynomial rings can be found computationally and explicitly.     

On Skew Triangular Matrix Rings

For a ring R, endomorphism α of R and positive integer n we define a skew triangular matrix ring T _n (R, α). By using an ideal theory of a skew triangular matrix ring T _n (R, α) we can determine prime, primitive, maximal ideals and radicals of the ring R[x; α]/ ? x ⁿ  ?, for each positive integer n, where R[x; α] is the skew polynomial ring, and ? x ⁿ  ? is the ideal generated by x ⁿ .     

Radicals of Ore Extension of Skew Armendariz Rings

Let R be a ring with an endomorphism α and an α-derivation δ. In this article, for a skew-Armendariz ring R we study some properties of skew polynomial ring R[x; α, δ]. In particular, among other results, we show that for an (α, δ)-compatible skew-Armendariz ring R, γ(R[x; α, δ]) = γ(R)[x; α, δ] = Ni?_*(R)[x; α, δ], where γ is a radical in the class of radicals which includes the Wedderburn, lower nil, Levitzky, and upper nil radicals. We also show that several properties, including the symmetric, reversible, ZC_n, zip, and 2-primal property, transfer between R and the skew polynomial ring R[x; α, δ], in case R is (α, δ)-compatible skew-Armendariz. As a consequence we extend and unify several known results.     

Graded Steinberg algebras and partial actions

Given a graded ample Hausdorff groupoid, we realise its graded Steinberg algebra as a partial skew inverse semigroup ring. We use this to show that for a partial action of a discrete group on a locally compact Hausdorff topological space which is totally disconnected, the Steinberg algebra of the associated groupoid is graded isomorphic to the corresponding partial skew group ring. We show that there is a one-to-one correspondence between the open invariant subsets of the topological space and the graded ideals of the partial skew group ring. We also consider the algebraic version of the partial $C^{?}$-algebra of an abelian group and realise it as a partial skew group ring via a partial action of the group on a topological space. Applications to the theory of Leavitt path algebras are given.     

Partial Skew Polynomial Rings and Goldie Rings

We prove that if R is a semiprime ring and α is a partial action of an infinite cyclic group on R, then R is right Goldie if and only if R[x; α] is right Goldie if and only if R?x; α? is right Goldie, where R[x; α] (R?x; α?) denotes the partial skew (Laurent) polynomial ring over R. In addition, R?x; α? is semiprime while R[x; α] is not necessarily semiprime.     

Ideal-Symmetric and Semiprime Rings

Lambek extended the usual commutative ideal theory to ideals in noncommutative rings, calling an ideal A of a ring R symmetric if rst ∈ A implies rts ∈ A for r, s, t ∈ R. R is usually called symmetric if 0 is a symmetric ideal. This naturally gives rise to extending the study of symmetric ring property to the lattice of ideals. In the process, we introduce the concept of an ideal-symmetric ring. We first characterize the class of ideal-symmetric rings and show that this ideal-symmetric property is Morita invariant. We provide a method of constructing an ideal-symmetric ring (but not semiprime) from any given semiprime ring, noting that semiprime rings are ideal-symmetric. We investigate the structure of minimal ideal-symmetric rings completely, finding two kinds of basic forms of finite ideal-symmetric rings. It is also shown that the ideal-symmetric property can go up to right quotient rings in relation with regular elements. The polynomial ring R[x] over an ideal-symmetric ring R need not be ideal-symmetric, but it is shown that the factor ring R[x]/xⁿR[x] is ideal-symmetric over a semiprime ring R.     

Automorphism Group of q-Quantum Torus Lie Algebra with q a Root of Unity*

Let α be an automorphism of a ring R. We study the skew Armendariz of Laurent series type rings (α-LA rings), as a generalization of the standard Armendariz condition from polynomials to skew Laurent series. We study on the relationship between the Baerness and p.p. property of a ring R and these of the skew Laurent series ring R[[x, x ^?1; α]], in case R is an α-LA ring. Moreover, we prove that for an α-weakly rigid ring R, R[[x, x ^?1; α]] is a left p.q.-Baer ring if and only if R is left p.q.-Baer and every countable subset of S _?(R) has a generalized countable join in R. Various types of examples of α-LA rings are provided.     

When is a Crossed Product by a Twisted Partial Action Azumaya?

In this article, we discuss necessary and sufficient conditions for the crossed product S = R_α G by a twisted partial action α of a finite group G on a ring R to be separable over its center.     

Leavitt Path Algebras as Partial Skew Group Rings

We realize Leavitt path algebras as partial skew group rings and give new proofs, based on partial skew group ring theory of the Cuntz–Krieger uniqueness theorem and simplicity criteria for Leavitt path algebras.     

Crossed Products by Twisted Partial Actions: Separability,Semisimplicity, and Frobenius Properties

In this article, we are concerned with the crossed product S = R☆_α G by a twisted partial action α of a group G on a ring R. We give necessary and sufficient (resp., sufficient) conditions for S to be a separable (resp., Frobenius, semisimple) extension of R.