Search

In this paper we continue the study of various ring theoretic properties of Morita contexts. Necessary and sufficient conditions are obtained for a general Morita context or a trivial Morita context or a formal triangular matrix ring to satisfy a certain ring property which is among being Kasch, completely primary, quasi-duo, 2-primal, NI, semiprimitive, projective-free, etc. We also characterize when a general Morita context is weakly principally quasi-Baer or strongly right mininjective. [ABSTRACT FROM AUTHOR]

AbstractBy means of the maximal left ring of quotients, this paper proves that every n-Lie m-derivation on a triangular ring can be resolved into a sum of an extremal n-derivation and an n-additive central mapping. As applications, n-Lie m-derivations on some classical triangular rings are characterized. [ABSTRACT FROM AUTHOR]

It is well known that the full matrix ring over a skew-field is a simple ring. We generalize this theorem to the case of semirings. We characterize the case when the matrix semiring M n (S) , of all n × n matrices over a semiring S, is congruence-simple, provided that either S has a multiplicatively absorbing element or S is commutative and additively cancellative. [ABSTRACT FROM AUTHOR]

We introduce a novel framework for assessing the centrality of idempotents within a ring by presenting a general concept that assigns a degree of centrality. This approach aligns with the previously established notions of semicentral and q-central idempotents by Birkenmeier and Lam. Specifically, we define an idempotent e in a ring R to be n-central, where n is a positive integer, if [e, R]ne = 0, where [x, y] represents the additive commutator xy-yx. If every idempotent in a ring R is n-central, we refer to R as n-Abelian. Our study lays the groundwork by presenting foundational results that support this concept and examines key features of n-central idempotents essential for appropriately categorizing n-Abelian rings among various generalizations of Abelian rings introduced in prior literature. We provide examples of n-central idempotents that do not fall under the categories of semicentral or q-central. Furthermore, we demonstrate that the ring of upper matrices Tn(R), where R is Abelian, is an n-abelian. We also prove that a ring where all of its idempotents are n-central is an exchange ring if and only if the ring is clean. [ABSTRACT FROM AUTHOR]

We construct examples of bounded below, noncontractible, acyclic complexes of finitely generated projective modules over some rings S, as well as bounded above, noncontractible, acyclic complexes of injective modules. The rings S are certain rings of infinite matrices with entries in the rings of commutative polynomials or formal power series in infinitely many variables. In the world of comodules or contramodules over coalgebras over fields, similar examples exist over the cocommutative symmetric coalgebra of an infinite-dimensional vector space. A simpler, universal example of a bounded below, noncontractible, acyclic complex of free modules with one generator, communicated to the author by Canonaco, is included at the end of the paper. [ABSTRACT FROM AUTHOR]

In this paper, we extend our investigation about the generalized comaximal graph introduced in Biswas et al. (Discrete Math Algorithms Appl 11(1):1950013, 2019a). The generalized comaximal graph is defined as follows: given a finite commutative ring R, the generalized comaximal graph G(R) is an undirected graph with its vertex set comprising elements of R and two distinct vertices u, v are adjacent if and only if there exists a non-zero idempotent e ∈ R such that u R + v R = e R . In this study, we focus on identifying the rings R for which the graph G(R) exhibits planarity. Moreover, we provide a characterization of the class of ring for which G(R) is toroidal, denoted by γ (G (R)) = 1 . Furthermore, we also evaluate the energy of the graph G(R). Finally, we demonstrate that the graph G(R) is always Hamiltonian for any finite commutative ring R. [ABSTRACT FROM AUTHOR]

Let A and B be rings and T = ( A M 0 B ) with M an A-B-bimodule. We first construct a semi-complete duality pair T over T using duality pairs over A and B, respectively. Then we characterize when a left T-module is Gorenstein projective, injective or flat with respect to the duality pair T . As applications, we investigate when a left T-module is projectively coresolved Gorenstein flat, Ding injective or Gorenstein flat. In addition, we give an estimate of Gorenstein flat dimension of a left T-module with respect to T . Finally, we construct model structures and recollements relative to these generalized Gorenstein modules. [ABSTRACT FROM AUTHOR]

We study the (z,k)-equivalence of the pairs of matrices over the ring of Gaussian integers and their reducibility to standard forms. It is shown that the number of standard forms of the pairs of matrices over this ring is finite. The classes of the pairs of matrices with minimum and maximum numbers of standard forms are presented. [ABSTRACT FROM AUTHOR]

Ring structures have been widely used in engineering devices and systems. When oscillations occur, the oscillation effects can be reduced by applying designable carbon nanotube (CNT)-reinforced materials to the ring. The dynamic characterizations and microscopic behaviors of CNT-reinforced rings of three distribution types of CNT, namely, X-distribution (X) type, uniform distribution (UD) type and O-distribution (O) type, are given. The influence of doped CNTs on the mechanical properties of the matrix ring is analyzed by comparing different CNT distributions on the membrane stiffness, bending stiffness and frequency. The microscopic membrane force-induced component and bending moment-induced component are studied to reflect the transverse and circumferential actions. The effects of different CNTs distributions on the transverse and circumferential action are evaluated. Analytical results demonstrate that the CNTs can improve the mechanical properties of the ring. The X type greatly affects the bending behavior, while the O type mainly affects the membrane behavior. Thus, for transverse oscillations dominated by the bending moment-induced component, the X type CNTs can greatly reduce the transverse oscillations influences. [ABSTRACT FROM AUTHOR]

Let 퐴 and 퐵 be rings and 푈 a (B , A) -bimodule. The Gorenstein projective, injective and flat complexes over the formal triangular matrix ring T = ( A 0 U B ) are explicitly described. These statements extend some earlier results of Enochs, Cortés-Izurdiaga, Torrecillas, Zhang and Mao in this direction. [ABSTRACT FROM AUTHOR]

On the Waring's problem for matrices over a commutative ring, there are some trace conditions provided for matrices eligibly expressed as sums of kth powers with $ k\in \{2,3,4,5,6,7,8\} $ k ∈ { 2 , 3 , 4 , 5 , 6 , 7 , 8 } in the literature. In this paper, we provide similar conditions for matrices written as sums of kth powers with $ k\in \{9,10,11,12,13,14,15,16\} $ k ∈ { 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 }. [ABSTRACT FROM AUTHOR]

Consider a graph on the non-singular matrices over a finite field, in which two distinct non-singular matrices are joined by an edge whenever their sum is singular. We prove an upper bound for the independence number of this graph. As a consequence, we obtain a lower bound for its chromatic number that significantly improves a previous result of Tomon. [ABSTRACT FROM AUTHOR]

This paper investigates relative ring theoretical properties in the context of formal triangular matrix rings. The first aim is to study the semiregularity of formal triangular matrix rings relative to an ideal. We prove that the formal triangular matrix ring T is T'-semiregular if and only if A is Isemiregular, B is K-semiregular and N = M for an ideal T' = (I0NK) of T = (A0MB). We also discuss the relative semiperfect formal triangular matrix rings in relation to the strong lifting property of ideals. Moreover, we have considered the behavior of relative semipotent and potent property of formal triangular matrix rings. Several examples are provided throughout the paper in order to highlight our results. [ABSTRACT FROM AUTHOR]

In this paper, we study the distribution of the cokernel of a general random Hermitian matrix over the ring of integers \mathcal {O} of a quadratic extension K of \mathbb {Q}_p. For each positive integer n, let X_n be a random n \times n Hermitian matrix over \mathcal {O} whose upper triangular entries are independent and their reductions are not too concentrated on certain values. We show that the distribution of the cokernel of X_n always converges to the same distribution which does not depend on the choices of X_n as n \rightarrow \infty and provide an explicit formula for the limiting distribution. This answers Open Problem 3.16 from the ICM 2022 lecture note of Wood [ Probability theory for random groups arising in number theory , 2022] in the case of the ring of integers of a quadratic extension of \mathbb {Q}_p. [ABSTRACT FROM AUTHOR]

6061 aluminum composites with 0.5 and 1 vol. % graphene nanoplatelets as well as 1 and 2 vol. % activated nanocarbon were manufactured by a powder metallurgy method. Scanning electron microscopy and Raman spectroscopy were used to study the morphology, structure, and distribution of nanocarbon reinforcements in the composite samples. Density Functional Theory (DFT) calculations were performed to understand the aluminum-carbon bonding and the effects of hybridized networks of carbon atoms on nanocarbon aluminum matrix composites. Scanning electron microscopy showed the good distribution and low agglomeration tendencies of nanoparticles in the composites. The formation of secondary phases at the materials interface was not detected in the hot-pressed composites. Raman spectroscopy showed structural changes in the reinforced composites after the manufacturing process. The results from Density Functional Theory calculations suggest that it is thermodynamically possible to form carbon rings in the aluminum matrix, which may be responsible for the improved mechanical strength. Our results also suggest that these carbon networks are graphene-like, which also agrees with the Raman spectroscopy data. Micro-Vickers hardness and compressive tests were used to determine the mechanical properties of the samples. Composites presented enhanced hardness, yield and ultimate strength compared to the 6061 aluminum alloy with no nanocarbon reinforcement. Ductility was also affected, as shown by the reduction in elongation and by the number of dimples in the fractured surfaces of the materials. [ABSTRACT FROM AUTHOR]

This paper studies the structure of semisimple rings using techniques of reverse mathematics, where a ring is left semisimple if the left regular module is a finite direct sum of simple submodules. The structure theorem of left semisimple rings, also called Wedderburn-Artin Theorem, is a famous theorem in noncommutative algebra, says that a ring is left semisimple if and only if it is isomorphic to a finite direct product of matrix rings over division rings. We provide a proof for the theorem in R C A 0 , showing the structure theorem for computable semisimple rings. The decomposition of semisimple rings as finite direct products of matrix rings over division rings is unique. Based on an effective proof of the Jordan-Hölder Theorem for modules with composition series, we also provide an effective proof for the uniqueness of the matrix decomposition of semisimple rings in R C A 0 . [ABSTRACT FROM AUTHOR]

Introduction: Objectives were to assess how different techniques (including a novel ringless sectional matrix approach) affect students' restorative outcomes, and their preferences for and preparedness to clinically implement new techniques with the aim of introducing appropriate techniques to the undergraduate curriculum. Materials and Methods: Students performed two class II composite restorations in plastic teeth using two sectional matrix techniques (separating ring vs. without [ringless]), and two composite restorative techniques (incremental vs. injection‐moulded bulk‐fill). Restorations were assessed on multiple parameters which were combined to rate them as "good" or not. Online surveys assessed students' preferences for and preparedness to clinically implement new techniques. Results: Contact area concavity (OR = 106, p <.001) and cervical marginal overhang (OR = 7.4, p <.001) were much more likely with the separating ring compared to the ringless sectional matrix technique. "Good" restorations were 29.5 times more likely when using ringless compared to separating ring techniques and 3.3 times more likely when using the injection‐moulding bulk‐fill compared to layered composite technique. A majority of students preferred the ringless to separating ring sectional matrix technique and the injection‐moulding bulk‐fill to layered composite technique. Large majorities felt prepared to implement the new techniques clinically with no or minimal guidance. Conclusions: A classic sectional matrix technique with separating ring resulted in a much greater occurrence of contact area concavity and cervical marginal overhang than a novel ringless approach. When allied with student preferences and clinical preparedness, inclusion of the novel ringless approach in the undergraduate curriculum can be supported alongside bulk‐fill injection‐moulding techniques. [ABSTRACT FROM AUTHOR]

The objective of this paper is to study for the first time the concept of square real matrices with symbolic 2-plithogenic split-complex entries. Many of their algebraic properties will be discussed and handled, where we find the formula of computing inverses, exponents, and powers of these matrices by building a ring isomorphism between the ring of split-complex symbolic 2-plithogenic matrices and the direct product of the symbolic 2-plithogenic matrices with itself. Also, we give the interested reader many related examples to clarify the validity of our work. [ABSTRACT FROM AUTHOR]

Let R be a ring with an endomorphism σ , F ∪ { 0 } the free monoid generated by U = { u 1 , ... , u t } with 0 added, and M a factor of F obtained by setting certain monomials in F to 0 such that M n = 0 for some n. Then we can form the non-semiprime skew monoid ring R [ M ; σ ]. A local ring R is called bleached if for any j ∈ J (R) and any u ∈ U (R) , the abelian group endomorphisms l u − r j and l j − r u of R are surjective. Using R [ M ; σ ] , we provide various classes of both bleached and non-bleached local rings. One of the main problems concerning strongly clean rings is to characterize the rings R for which the matrix ring M n (R) is strongly clean. We investigate the strong cleanness of the full matrix rings over the skew monoid ring R [ M ; σ ]. [ABSTRACT FROM AUTHOR]

Let G be a group, let A = ⊕ g ∈ G A g be an epsilon-strongly graded ring over G, let R := A 1 be the homogeneous component associated with the identity of G, and let 홿횒회횂 ⁢ (R) be the Picard semigroup of R. In the first part of this paper, we prove that the isomorphism class [ A g ] is an element of 홿횒회횂 ⁢ (R) for all g ∈ G . Moreover, the association g ↦ [ A g ] determines a partial representation of G on 홿횒회횂 ⁢ (R) which induces a partial action γ of G on the center Z ⁢ (R) of R. Sufficient conditions for A to be an Azumaya R γ -algebra are presented if R is commutative. In the second part, we study when B is a partial crossed product in the following cases: B = M n ⁡ (A) is the ring of matrices with entries in A, or B = END A (M) = ⊕ l ∈ G Mor A (M , M) l is the direct sum of graded endomorphisms of graded left A-modules M with degree l, or B = END A ⁡ (M) where M = A ⊗ R N is the induced module of a left R-module N. Assuming that R is semiperfect, we prove that there exists a subring of A which is an epsilon-strongly graded ring over a subgroup of G and it is graded equivalent to a partial crossed product. [ABSTRACT FROM AUTHOR]

A ring R is called semiclean if every element of R can be expressed as sum of a periodic element and a unit. In this paper, we introduce a new class of ring, which is the *-version of the semiclean ring, i.e. the *-semiclean ring. A *-ring is *-semiclean if each element is a sum of a *-periodic element and a unit. The term *-semiclean is a stronger notion than semiclean. In this paper, many properties of *-semiclean rings are discussed. It is proved that if p ∈ P(R) such that pRp and (1 - p)R(1 - p) are *-semiclean rings, then R is also a *-semiclean ring. As a result, the matrix ring Mn(R) over a *-semiclean ring is *-semiclean. A characterization that when the group rings RCr and RG are *-semiclean is done, where R is a finite commutative local ring, Cr is a cyclic group of order r, and G is a locally finite abelian group. We have also found sufficient conditions when the group rings RC3, RC4, RQ8, and RQ2n are *-semiclean, where R is a commutative local ring. We have also demonstrated that the group ring Z2D6 is a *-semiclean ring (which is not a *-clean ring). [ABSTRACT FROM AUTHOR]

This paper is devoted to introducing and studying two concepts, σ-skew strongly Mreflexive and σ-skew strongly M-nil-reflexive on monoid rings, which are generalizations of strongly M-reflexive and M-compatible. The paper covers the basic properties of skew monoid rings of the form R∗M. It is shown that if R is a left AP P (quasi Armendariz, semiprime rings, respectively), then R is σ-skew strongly M-reflexive. Moreover, if R is a NI-ring and M is a u.p-monoid, then R is σ-skew strongly M-nil-reflexive. Additionally, under some necessary and sufficient conditions, a skew monoid ring R ∗ M is proven to be σ-skew strongly M-nil-reflexive when σ : M → Aut(R) is a monoid homomorphism. Furthermore, if R is a left AP P, then the upper triangular matrix ring Tn(R) is σ¯-skew strongly M-nil-reflexive, where n is a positive integer. Finally, the paper provides some examples and discusses related results from the subject. [ABSTRACT FROM AUTHOR]

A ring R is said to be n-fine if every nonzero element in R can be written as a sum of a nilpotent and n units in R. The class of these rings contains fine rings and n-good rings in which each element is a sum of n units. Fundamental properties of such rings are obtained. One of the main results of this paper is that the m × m matrix ring Mm(R) over any arbitrary ring R is 2-fine. Furthermore, the m × m matrix ring Mm(R) over a n-fine ring R is n-fine. [ABSTRACT FROM AUTHOR]

In this paper, the author gives a description of the derivations of U T M n (R) , the ring of upper triangular matrices over an associative ring R with identity. The main result states that if D is an arbitrary derivation of the ring U T M n (R) and A ∈ U T M n (R) , then there are matrices, such that the derivative D (A) is a linear combination of the values of well-known derivations of these matrices. [ABSTRACT FROM AUTHOR]

Automorphism groups of formal matrix algebras with zero trace ideals are studied. Such an algebra is represented as a splitting extension of some ring by some nilpotent ideal. Using this extension, the study of the structure of the automorphism group of an algebra in a certain sense is reduced to the study of the structure of some its subgroups and quotient groups. Then the structure of these subgroups and factor groups is found. The case of formal triangular matrix rings is specially considered. At the end of the paper we study automorphisms of ordinary matrix rings. In this case, a well-known homomorphism from the group of outer automorphisms to the Picard group of a certain ring is used. [ABSTRACT FROM AUTHOR]

Let A and B be unital rings. An additive map T : A → B is called a weighted Jordan homomorphism if c = T (1) is an invertible central element and c T (x 2) = T (x) 2 for all x ∈ A. We provide assumptions, which are in particular fulfilled when A = B = M n (R) with n ≥ 2 and R any unital ring with 1 2 , under which every surjective additive map T : A → B with the property that T (x) T (y) + T (y) T (x) = 0 whenever xy = yx = 0 is a weighted Jordan homomorphism. Further, we show that if A is a prime ring with char (A) ≠ 2 , 3 , 5 , then a bijective additive map T : A → A is a weighted Jordan homomorphism provided that there exists an additive map S : A → A such that S (x 2) = T (x) 2 for all x ∈ A. [ABSTRACT FROM AUTHOR]

Measuring neutron capture cross sections of radioactive nuclei is a crucial step towards a better understanding of the origin of the elements heavier than iron. For decades, the precise measurement of direct neutron capture cross sections in the "stellar" energy range (eV up to a few MeV) was limited to stable and longer-lived nuclei that could be provided as physical samples and then irradiated with neutrons. New experimental methods are now being developed to extend these direct measurements towards shorter-lived radioactive nuclei ( t 1 / 2 < 1 y). One project in this direction is a low-energy heavy-ion storage ring coupled to the ISAC facility at TRIUMF, Canada's accelerator laboratory in Vancouver BC, which has a compact neutron source in the ring matrix. Such a pioneering facility could be built within the next 10 years and store a wide range of radioactive ions provided directly from the existing ISOL facility, allowing for the first time to carry out direct neutron capture measurements on short-lived isotopes in inverse kinematics. [ABSTRACT FROM AUTHOR]

The differential Brauer monoid of a differential commutative ring R is defined. Its elements are the isomorphism classes of differential Azumaya R algebras with operation from tensor product subject to the relation that two such algebras are equivalent if matrix algebras over them, with entry-wise differentiation, are differentially isomorphic. The fine Bauer monoid, which is a group, is the same thing without the differential requirement. The differential Brauer monoid is then determined from the fine Brauer monoids of R and R^D and the submonoid of the Brauer monoid whose underlying Azumaya algebras are matrix rings. [ABSTRACT FROM AUTHOR]

For a matrix incidence algebra over a field, there is a full description of its generating subsets. We generalise this result to the case of matrix incidence rings over an arbitrary unital associative ring. Generating subsets of the following type are under consideration: if one adds all scalar matrices to the subset, then this greater set will generate the incidence ring. We obtain the full characterisation of such subsets. Moreover, their minimum cardinality is calculated precisely when the ring of coefficients is simple or semilocal. In order to obtain these results we introduce the graph of full differences of a ring and study its clique number. [ABSTRACT FROM AUTHOR]

We study the singular ideal, the Jacobson radical and the maximal quotient rings of the matrix wreath product of two algebras and certain subrings as defined in (Alamhadi-Jain-Alsulami and Zelmanov, Electron. Res. Announc. Math. Sci. (2017)) [1]. We give formulas for computing the singular ideal and the Jacobson radical of a matrix wreath product. We also give the structure of the maximal right quotient ring of the infinite matrix ring of a right nonsingular algebra and, as consequence, we give the maximal quotient ring of a matrix wreath product. As an application of the fact that matrix wreath product of nonsingular algebras is nonsingular we construct another algebra (similar to the one in [3]) showing that any nonsingular algebra of countable dimension can be embedded in a finitely generated nonsingular algebra. [ABSTRACT FROM AUTHOR]

We consider the isomorphism problem for formal matrix rings over a given ring. Principal factor matrices of such rings play an important role in this case. [ABSTRACT FROM AUTHOR]

We study the Poincaré series of the mixed and pure trace rings of generic matrices. These series are known to be rational functions. We obtain an explicit formula in lowest terms in the case of 2 × 2 matrices; a denominator, which we presume but have not been able to prove to be in lowest terms, in the case of 3 × 3 matrices; and a conjectured denominator in the general case. [ABSTRACT FROM AUTHOR]

We present a non-commutative algorithm for the multiplication of a 2 × 2 block-matrix by its adjoint, defined by a matrix ring anti-homomorphism. This algorithm uses 5 block products (3 recursive calls and 2 general products). It works unconditionally in positive characteristic, or over the complex field with transposition, but for instance not over the complex field with conjugate-transpose. The resulting algorithm for arbitrary dimensions is a reduction of multiplication of a matrix by its adjoint to general matrix product, improving by a constant factor previously known reductions. We prove also that there is no algorithm derived from bilinear forms using only four products and the adjoint of one of them. Second we give novel dedicated algorithms for the complex field and the quaternions to compute the multiplication alternatively, taking advantage of the structure of the involved matrix-polynomial arithmetic. We then analyze the respective ranges of predominance of the two strategies. Finally we propose schedules with low memory footprint that support a fast and memory efficient practical implementation over a prime field. [ABSTRACT FROM AUTHOR]

The object of this article is to initiate the study of a class of rings in which the right duo property is applied in relation to powers of elements and the monoid of all regular elements. Such rings shall be called right exp-DR. We investigate the structures of group rings, right quotient rings, matrix rings and (skew) polynomial rings, through the study of right exp-DR rings. In addition, we provide a method of constructing finite non-abelian p -groups for any prime p. [ABSTRACT FROM AUTHOR]

The classical Morita Theorem for rings established the equivalence of three statements involving categorical equivalences, isomorphisms between corners of finite matrix rings, and bimodule homomorphisms. A fourth equivalent statement (established later) involves an isomorphism between infinite matrix rings. In our main result, we establish the equivalence of analogous statements involving graded categorical equivalences, graded isomorphisms between corners of finite matrix rings, graded bimodule homomorphisms, and graded isomorphisms between infinite matrix rings. [ABSTRACT FROM AUTHOR]

In this work, we study codes generated by elements that come from group matrix rings. We present a matrix construction which we use to generate codes in two different ambient spaces: the matrix ring M k (R) and the ring R, where R is the commutative Frobenius ring. We show that codes over the ring M k (R) are one sided ideals in the group matrix ring M k (R) G and the corresponding codes over the ring R are G k -codes of length kn. Additionally, we give a generator matrix for self-dual codes, which consist of the mentioned above matrix construction. We employ this generator matrix to search for binary self-dual codes with parameters [72, 36, 12] and find new singly-even and doubly-even codes of this type. In particular, we construct 16 new Type I and 4 new Type II binary [72, 36, 12] self-dual codes. [ABSTRACT FROM AUTHOR]

We prove that a ring R is an $n \times n$ matrix ring (that is, $R \cong \mathbb {M}_n(S)$ for some ring S) if and only if there exists a (von Neumann) regular element x in R such that $l_R(x) = R{x^{n-1}}$. As applications, we prove some new results, strengthen some known results and provide easier proofs of other results. For instance, we prove that if a ring R has elements x and y such that $x^n = 0$ , $Rx+Ry = R$ and $Ry \cap l_{R}(x^{n-1}) = 0$ , then R is an $n \times n$ matrix ring. This improves a result of Fuchs ['A characterisation result for matrix rings', Bull. Aust. Math. Soc. 43 (1991), 265–267] where it is proved assuming further that the element y is nilpotent of index two and $x+y$ is a unit. For an ideal I of a ring R , we prove that the ring $(\begin {smallmatrix} R & I \\ R & R \end {smallmatrix})$ is a $2 \times 2$ matrix ring if and only if $R/I$ is so. [ABSTRACT FROM AUTHOR]

Consider ℜ as a prime ring which is non-commutative in structure with a suitable characteristic. Here, 풵 ⁢ (ℜ) is the center of ℜ and 풬 is the Utumi ring of quotients where 풞 is the extended centroid of ℜ . Suppose 풫 to be a Lie ideal of ℜ which is non-central. Let 풦 be a generalized derivation of ℜ related with derivation μ of ℜ . If 풦 satisfies certain typical algebraic identities, then we prove that 풦 is either the identity map or the zero map or the scalar map and further information is also drawn on the associated scalar unless ℜ embeds in M 2 ⁢ (풞) , a matrix ring of order 2 × 2 over 풞 . [ABSTRACT FROM AUTHOR]

A regular graph of the ring of n × n matrices over a field is a graph whose vertices are nonsingular matrices. Two different matrices are adjacent if their sum is singular. In 2009, S. Akbari, M. Jamaali, and S. Seed Fakhari found that the clique number of this graph is finite whenever the field is not of characteristic 2. The same authors asked if the chromatic number of the graph is finite (for fields of characteristic 0 this question is still open). In this paper, we introduce a concept of total and regular graph of a polynomial, generalizing the regular graph of a matrix ring. We investigate some properties of these graphs and their relationship with the above question. Several new open questions are also posed. [ABSTRACT FROM AUTHOR]

In this paper, we prove that every multiplicative anti-derivation map of an arbitrary generalized n-matrix ring is additive. Even more, as consequence, we get the same conclusion for the class of triangular n-matrix rings, unital prime rings with a nontrivial idempotent, standard operator algebras and factor von Neumann algebras. [ABSTRACT FROM AUTHOR]

The present work aims to exploit the interplay between the algebraic properties of rings and the graph-theoretic structures of some associated graphs. We introduce commutatively closed graphs and investigate properties of commutatively closed subsets of a ring with the help of graph theory. In particular, we compute the diameter of matrix rings and Artinian semisimple algebras. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring, m>1 an integer and q>1 an odd number. The following questions are addressed: (1) when is every element in M n (R) and, respectively, in T n (R) a product of q-potents? (2) when is every element in M n (R) and, respectively, in T n (R) a product of mq-potents? and (3) when is every element in M n (R) and, respectively, in T n (R) a product of an idempotent and a q-potent? [ABSTRACT FROM AUTHOR]

Let A 1 , ... , A s be unitary commutative rings which do not have non-trivial idempotents and let A = A 1 ⊕ ⋯ ⊕ A s be their direct sum. We describe all idempotents in the 2 × 2 matrix ring M 2 (A [ [ X ] ]) over the ring A [ [ X ] ] of formal power series with coefficients in A and in an arbitrary set of variables X. We apply this result to the matrix ring M 2 ( Z n [ [ X ] ]) over the ring Z n [ [ X ] ] where Z n ≅ Z / n Z for an arbitrary positive integer n greater than 1. Our proof is elementary and uses only the Cayley-Hamilton theorem (for 2 × 2 matrices only) and, in the special case A = Z n , the Chinese remainder theorem and the Euler-Fermat theorem. [ABSTRACT FROM AUTHOR]

A ring R is called filial (left weakly filial) if it satisfies the following condition: for all subrings J and I of R, if J is an ideal of I and I is an ideal of R, then J is an ideal (left ideal) of R. If every subring of R is filial (left weakly filial), then R is called fully filial (fully left weakly filial). In this paper, some new characterizations of left weakly filial rings in terms of the prime radical are given. Left weakly filial rings that are matrix rings, polynomial rings and the direct sum of copies of a ring are also investigated. Recently, in [3], [4] and [13], classification theorems for semiprime torsion fully filial rings and for torsion-free fully filial rings were presented. We study fully left weakly filial rings and show that semiprime torsion rings and semiprime torsion-free rings are fully left weakly filial if and only if they are fully filial. [ABSTRACT FROM AUTHOR]