Noncommutative Rings Research Articles

On total directed graphs of non-commutative rings

For a non-commutative ring R, the left total directed graph of R is a directed graph with vertex set as R and for the vertices x and y, x is adjacent to y if and only if there is a non-zero r∈R which is different from x and y, such that rx+yr is a left zero-divisor of R. In this paper, we discuss some very basic results of left (as well as right) total directed graph of R. We also study the coloring of left total directed graph of R directed graph.

Generalized Łukasiewicz rings

This paper extends the study of commutative rings whose ideals form an MV-algebra as carried out by Belluce and Di Nola (Math Log Q 55(5):468---486, 2009) to non-commutative rings. We study and characterize all rings whose ideals form a pseudo-MV-algebra, which shall be called here generalized źukasiewicz rings. We obtain that up to isomorphism, these are exactly the direct sums of unitary special primary rings.

Jordan [! \large !]$\sigma $-derivations of prime rings

Let $R$ be a noncommutative prime ring with extended centroid~$C$ and with $Q_{mr}(R)$ its maximal right ring of quotients. From the viewpoint of functional identities, we give a complete characterization of Jordan $\sigma $-derivations of $R$ with $\sigma $ an epimorphism. Precisely, given such a Jordan $\sigma $-derivation $\de \colon R\to Q_{mr}(R)$, it is proved that either $\delta $ is a $\sigma $-derivation or a derivation $d\colon R\to Q_{mr}(R)$ and a unit $u\in Q_{mr}(R)$ exist such that $\delta (x)=ud(x)+\mu (x)u$ for all $x\in R$, where $\mu \colon R\to C$ is an additive map satisfying $\mu (x^2)=0$ for all $x\in R$. In addition, if $\sigma $ is an X-outer automorphism, then $\delta $ is always a $\sigma $-derivation.

On zero-divisor graphs of skew polynomial rings over non-commutative rings

In this paper, we continue to study zero-divisor properties of skew polynomial rings [Formula: see text], where [Formula: see text] is an associative ring equipped with an endomorphism [Formula: see text] and an [Formula: see text]-derivation [Formula: see text]. For an associative ring [Formula: see text], the undirected zero-divisor graph of [Formula: see text] is the graph [Formula: see text] such that the vertices of [Formula: see text] are all the nonzero zero-divisors of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are connected by an edge if and only if [Formula: see text] or [Formula: see text]. As an application of reversible rings, we investigate the interplay between the ring-theoretical properties of a skew polynomial ring [Formula: see text] and the graph-theoretical properties of its zero-divisor graph [Formula: see text]. Our goal in this paper is to give a characterization of the possible diameters of [Formula: see text] in terms of the diameter of [Formula: see text], when the base ring [Formula: see text] is reversible and also have the [Formula: see text]-compatible property. We also completely describe the associative rings all of whose zero-divisor graphs of skew polynomials are complete.

Divided powers and composition in integro-differential algebras

An integro-differential algebra of arbitrary characteristic is given the structure of a uniform topological space, such that the ring operations as well as the derivation (= differentiation operator) and Rota–Baxter operator (= integral operator) are uniformly continuous. Using topological techniques and the central notion of divided powers, this allows one to introduce a composition for (topologically) complete integro-differential algebras; this generalizes the series case, viz. meaning formal power series in characteristic zero and Hurwitz series in positive characteristic. The canonical Hausdorff completion for pseudometric spaces is shown to produce complete integro-differential algebras.The setting of complete integro-differential algebras allows us to describe exponential and logarithmic elements in a way that reflects the “integro-differential properties” known from analysis. Finally, we prove also that any complete integro-differential algebra is saturated, in the sense that every (monic) linear differential equation possesses a regular fundamental system of solutions.While the paper focuses on the commutative case, many results are given for the general case of (possibly noncommutative) rings, whenever this does not require substantial modifications.

Zero commutativity of nilpotent elements skewed by ring endomorphisms

ABSTRACTThe reversible property is an important role in noncommutative ring theory. Recently, the study of the reversible ring property on nilpotent elements is established by Abdul-Jabbar et al., introducing the concept of commutativity of nilpotent elements at zero (simply, a CNZ ring) as a generalization of reversible rings. We here study this property skewed by a ring endomorphism α, and such ring is called a right α-skew CNZ ring which is an extension of CNZ rings as well as a generalization of right α-skew reversible rings, and then investigate the structure of right α-skew CNZ rings and their related properties. Consequently, several known results are obtained as corollaries of our results.

When is a power of the Frobenius map on a noncommutative ring a homomorphism?

Let φ(x)=xp be the Frobenius map on an associative unital ring R with prime characteristic p. It is well-known that, whenever R is commutative, φn is a ring homomorphism, for all positive integers n. The converse, however, is not true in general. Indeed, we prove that, if R is m-Engel, for some positive integer m, then there exists a positive integer n0 depending only on m such that, for all n≥n0, φn is a ring homomorphism with central image. Conversely, if any one of the following conditions holds: φn respects addition, φn respects multiplication, φn respects Lie multiplication, or the image of φn is commutative, then R is m-Engel, for some m depending only on n. Consequently, if φ is surjective, and any one of the aforementioned conditions holds, then R must be commutative.

The Wedderburn–Artin Theorem for Paragraded Rings

In this paper, we prove the paragraded version of the Wedderburn–Artin theorem. Following the methods known from the abstract case, we first prove the density theorem and observe the matrix rings whose entries are from a paragraded ring. However, in order to arrive at the desired structure theorem, we introduce the notion of a Jacobson radical of a paragraded ring and prove some properties which are analogous to the abstract case. In the process, we study the faithful and irreducible paragraded modules over noncommutative paragraded rings and prove the paragraded version of the well-known Schur lemma.

Reversibles elements in rings

We study some properties related to zero divisors and reversibility in noncommutative rings.

Twisted bialgebroids versus bialgebroids from a Drinfeld twist

Bialgebroids (respectively Hopf algebroids) are bialgebras (Hopf algebras) over noncommutative rings. Drinfeld twist techniques are particularly useful in the (deformation) quantization of Lie algebras as well as the underlying module algebras (=quantum spaces). A smash product construction combines both of them into the new algebra which, in fact, does not depend on the twist. However, we can turn it into a bialgebroid in a twist-dependent way. Alternatively, one can use Drinfeld twist techniques in a category of bialgebroids. We show that both the techniques indicated in the title—the twisting of a bialgebroid or constructing a bialgebroid from the twisted bialgebra—give rise to the same result in the case of a normalized cocycle twist. This can be useful for the better description of a quantum deformed phase space. We argue that within this bialgebroid framework one can justify the use of deformed coordinates (i.e. spacetime noncommutativity), which are frequently postulated in order to explain quantum gravity effects.

Commuting maps on rank-1 matrices over noncommutative division rings

ABSTRACTLet n≥3 be a natural number. Let Mn(𝔻) be the ring of all n×n matrices over a noncommutative division ring 𝔻. In the present paper, we will find the description of all additive mappings such that [G(y),y] = G(y)y−yG(y) = 0 for all rank-1 matrix y. Precisely, we will prove that G(x) = λx+μ(x) for all x∈Mn(𝔻), where λ lies in the center of 𝔻 and μ is a central map.

Some noncommutative rings constructed on the base of polynomials x2 + tx + 1 and their zero divisors

Some noncommutative rings constructed on the base of polynomials x2 + tx + 1 and their zero divisors

Coherent rings, fp-injective modules, dualizing complexes, and covariant Serre–Grothendieck duality

For a left coherent ring A with every left ideal having a countable set of generators, we show that the coderived category of left A-modules is compactly generated by the bounded derived category of finitely presented left A-modules (reproducing a particular case of a recent result of Stovicek with our methods). Furthermore, we present the definition of a dualizing complex of fp-injective modules over a pair of noncommutative coherent rings A and B, and construct an equivalence between the coderived category of A-modules and the contraderived category of B-modules. Finally, we define the notion of a relative dualizing complex of bimodules for a pair of noncommutative ring homomorphisms A \to R and B \to S, and obtain an equivalence between the R/A-semicoderived category of R-modules and the S/B-semicontraderived category of S-modules. For a homomorphism of commutative rings A\to R, we also construct a tensor structure on the R/A-semicoderived category of R-modules. A vision of semi-infinite algebraic geometry is discussed in the introduction.

On noncommutative piecewise Noetherian rings

ABSTRACTWe extend the definition of a piecewise Noetherian ring to the noncommutative case, and investigate various properties of such rings. In particular, we show that a ring with Krull dimension is piecewise Noetherian. Certain fully bounded piecewise Noetherian rings have Gabriel dimension and exhibit the Gabriel correspondence between prime ideals and indecomposable injective modules.

A basic functional identity with applications to Jordan σ-biderivations

ABSTRACTLet R be a noncommutative prime ring with extended centroid C and maximal left ring of quotients Qml(R). The aim of the paper is to study a basic functional identity concerning bi-additive maps on R. Precisely, it is proved that a bi-additive map B:R×R→Qml(R) satisfying [B(x,y),[x,y]] = 0 for all x,y∈R must be of the form (x,y)↦λ[x,y]+μ(x,y) for x,y∈R, where λ∈C and μ:R×R→C is a bi-additive map. As applications to the theorem, Jordan σ-biderivations with σ an epimorphism and additive commuting maps on noncommutative Lie ideals of R are characterized.

Reflexivity with maximal ideal axes

ABSTRACTThe reflexive property for ideals was introduced by Mason and has important roles in noncommutative ring theory. We in this note study rings with the reflexivity whose axis is given by maximal ideals (simply, an RM ring) which are a generalization of symmetric rings. It is first shown that the reflexivity of a ring and the RM ring property are independent of each other, noting that both of them are generalizations of ideal-symmetric rings. We connect RM rings with reflexive rings in various situations raised naturally in the procedure. As a generalization of RM rings, we also study the structure of the reflexivity with the maximal ideal axis on idempotents (simply, an RMI ring) and then investigate the structure of minimal non-Abelian RMI rings (with or without identity) up to isomorphism.

Some algebras similar to the 2×2 Jordanian matrix algebra

ABSTRACTThe impetus for this study is the work of Dumas and Rigal on the Jordanian deformation of the ring of coordinate functions on 2×2 matrices. We are also motivated by current interest in birational equivalence of noncommutative rings. Recognizing the construction of the Jordanian matrix algebra as a skew polynomial ring, we construct a family of algebras relative to differential operator rings over a polynomial ring in one variable which are birationally equivalent to the Weyl algebra over a polynomial ring in two variables.

Realisation of linear time-varying systems

ABSTRACTIn this paper, the realisation problem of linear multi-input multi-output, time-varying systems is studied. The approach, based on the theory of non-commutative polynomial rings, yields explicit and simple formulas for computation of the state coordinates as well as for state equations in observable canonical form. The formulas are based on (left) Euclidean polynomial division.

Factorizations of block triangular matrices

Transfer homomorphisms have long been used as tools in factorization theory. The idea is to transfer information about factorizations in a semigroup of interest from simpler, easier-to-understand semigroups. Unfortunately there exist noncommutative semigroups which admit no transfer homomorphism to any commutative semigroup. The weak transfer homomorphism was introduced to study such semigroups. In particular, these weak transfer homomorphisms were used to study factorizations of semigroups of upper-triangular matrices with entries coming from a commutative integral domain. In this work we extend this study, finding weak transfer homomorphisms to both commutative and noncommutative semigroups from upper triangular matrices with entries coming from noncommutative rings. In particular, we study factorizations of block triangular matrices.

On Ore extension and skew power series rings with some restrictions on zero-divisors

The study of rings with right Property ([Formula: see text]), has done an important role in noncommutative ring theory. Following literature, a ring [Formula: see text] has right Property ([Formula: see text]) if every finitely generated two-sided ideal consisting entirely of left zero-divisors has a nonzero right annihilator. Our results in this paper concerns the right Property ([Formula: see text]) of Ore extensions as well as skew power series rings. We will show that if [Formula: see text] is a right duo ring, then the skew power series ring [Formula: see text] has right Property ([Formula: see text]), when [Formula: see text] is right Noetherian and [Formula: see text]-compatible. Moreover, for a right duo ring [Formula: see text] which is [Formula: see text]-compatible, it is shown that (i) the Ore extension ring [Formula: see text] has right Property ([Formula: see text]) and (ii) [Formula: see text] is right zip if and only if [Formula: see text] is right zip. As a corollary of our results, we provide answers to some open questions related to Property [Formula: see text], raised in [C. Y. Hong, N. K. Kim, Y. Lee and S. J. Ryu, Rings with Property ([Formula: see text]) and their extensions, J. Algebra 315 (2007) 612–628].