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Let RR be a unital associative ring. Our motivation is to prove that left derivations in column finite matrix rings over RR are equal to zero and demonstrate that a left derivation d:T→Td:{\mathcal{T}}\to {\mathcal{T}} in the infinite upper triangular matrix ring T{\mathcal{T}} is determined by left derivations dj{d}_{j} in R(j=1,2,…)R\left(j=1,2,\ldots ) satisfying d((aij))=(bij)d\left(\left({a}_{ij}))=\left({b}_{ij}) for any (aij)∈T\left({a}_{ij})\in {\mathcal{T}}, where bij=dj(a11),i=1,0,i≠1.{b}_{ij}=\left\{\begin{array}{ll}{d}_{j}\left({a}_{11}),& i=1,\\ 0,& i\ne 1.\end{array}\right. The similar results about Jordan left derivations are also obtained when RR is 2-torsion free.

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]

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]

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]

S. Frisch, showed that the integer-valued polynomials on upper triangular matrix ring Int T n (K) (T n (D)) : = { f ∈ T n (K) [ x ] | f (T n (D)) ⊆ T n (D) } is a ring, where D is an integral domain with field of fractions K. Let R 1 ⊆ R 2 be commutative rings with identity. In this paper, we study the set Int T n (R 2) (Ω , T n (R 1)) : = { f ∈ T n (R 2) [ x ] | f (Ω) ⊆ T n (R 1) } for some subsets Ω ⊆ T n (R 1) . We generalize Frisch's result and show that Int T n (R 2) (T n (R 1)) : = Int T n (R 2) (T n (R 1) , T n (R 1)) is a ring. We state a lower bound for the Krull dimension of the integer-valued polynomials on upper triangular matrix rings. Finally, we state the concept of Skolem closure of an ideal of the integer-valued polynomials on upper triangular matrix rings and as a consequence, we obtain a classification of maximal ideals of the integer-valued polynomials on upper triangular matrix rings. [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]

Let A be a unital Banach algebra and M be a unital A -bimodule. We show that additive mappings δ , τ : A → M satisfy the identity A − 1 δ (A) + δ (A) A − 1 + A τ (A − 1) + τ (A − 1) A = 0 for all invertible element A in A if and only if there exists a Jordan derivation D such that δ (A) = D (A) + A δ (I) and τ (A) = D (A) − δ (I) A for all A ∈ A . We also characterize additive mappings δ , τ : A → M satisfying the identity δ (A) B + A τ (B) = M for all A , B ∈ A with AB = N, where N is an invertible element in A and M is a fixed element in M. Similar conclusions are obtained for additive mappings from a matrix ring M n (D) into its unital bimodule satisfying any of the above identities, where D is a division ring and char D ≠ 2 . [ABSTRACT FROM AUTHOR]

AbstractThere are many ways for zero-dividing polynomials to go up to zero-dividing ideals when the base rings are IFP. The importance of these in ring theory leads us to consider the following ring conditions and study new useful roles of matrices for ring theory. Let R be a ring and a,b∈R\{0}. The first is the condition (*) that if ab = 0 then Ib = 0 for some nonzero ideal I⊆RaR of R or aJ = 0 for some nonzero ideal J⊆RbR of R. It is shown that from given any IFP ring, there can be constructed a non-IFP ring with the condition (*). We prove that a semiprime ring R with the condition (*) is both right and left nonsingular. The second is the condition (**) that if ab = 0 then IJ = 0 for some nonzero ideals I⊆RaR and J⊆RbR of R. We prove that every ring can be a subring of rings with the condition (**), that if R is an irreducible ring with the condition (**) then R is either a domain or non-semiprime, and that the condition (**) passes to polynomial rings when the base ring is semiprime. Various sorts of examples are given to illustrate and delimit the results obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study the combinatorial relations between the cokernels cok ⁡ (A n + p ⁢ x i ⁢ I n ) ( 1 ≤ i ≤ m ), where A n is an n × n matrix over the ring of p-adic integers ℤ p , I n is the n × n identity matrix and x 1 , ... , x m are elements of ℤ p whose reductions modulo p are distinct. For a positive integer m ≤ 4 and given x 1 , ... , x m ∈ ℤ p , we determine the set of m-tuples of finitely generated ℤ p -modules (H 1 , ... , H m) for which (cok ⁡ (A n + p ⁢ x 1 ⁢ I n ) , ... , cok ⁡ (A n + p ⁢ x m ⁢ I n )) = (H 1 , ... , H m) for some matrix A n . We also prove that if A n is an n × n Haar random matrix over ℤ p for each positive integer n, then the joint distribution of cok ⁡ (A n + p ⁢ x i ⁢ I n ) ( 1 ≤ i ≤ m ) converges as n → ∞ . [ABSTRACT FROM AUTHOR]

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]

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]

A ring R is said to be i-reversible if for every a , b ∈ R , a b is a nonzero idempotent implies b a is an idempotent. It is known that the rings M n (R) and T n (R) (the ring of all upper triangular matrices over R) are not i-reversible for n ≥ 3. In this paper, we provide a nontrivial i-reversible subring of M n (R) when n ≥ 3 and R has only trivial idempotents. We further provide a maximal i-reversible subring of T n (R) for each n ≥ 3 , if R is a field. We then give conditions for i-reversibility of trivial, Dorroh and Nagata extensions. Finally, we give some independent sufficient conditions for i-reversibility of polynomial rings, and more generally, of skew polynomial rings. [ABSTRACT FROM AUTHOR]

Which groups can occur as the group of units in a ring? Such groups are called realizable. Though the realizable members of several classes of groups have been determined (e.g., cyclic, odd order, alternating, symmetric, finite simple, indecomposable abelian, and dihedral), the question remains open. The general linear groups are realizable by definition: they are the units in the corresponding matrix rings. In this paper, we study the realizability of two closely related linear groups, the special linear groups and the affine general linear groups. We determine which special linear groups of degree 2 over a finite field are realizable by a finite ring, and we determine which affine general linear groups of degree 1 over a cyclic group are realizable by a finite ring. We also give partial results for certain linear groups of other degrees and for rings of characteristic zero. [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 D be a division ring such that either char(D) ≠ 2, 3 or D is not a field and char(D) ≠ 2. Let R = Mn(D) be the matrix ring over D, where n > 1. Let m, k be fixed invertible elements in R. The main purpose of the paper is to give a description of a bijective additive map f : R → R, satisfying the identity f(x)f(y) = m for every x, y ∈ R with xy = k, which gives a correct version of a result due to Catalano et al. in 2019. [ABSTRACT FROM AUTHOR]

We examine those matrix rings whose entries lie in periodic rings equipped with some additional properties. Specifically, we prove that the famous Diesl's question whether or not a ring R being nil-clean implies that the matrix ring M n (R) over R is also nil-clean for all n ≥ 1 is paralleling to the corresponding implication for (abelian, local) periodic rings. Besides, we study when the endomorphism ring E (G) of an abelian group G is periodic. Concretely, we establish that E (G) is periodic exactly when G is finite as well as we find a complete necessary and sufficient condition when the endomorphism ring over an abelian group is strongly m-nil clean for some natural number m thus refining an "old" result concerning strongly nil-clean endomorphism rings. Responding to a question when a group ring is periodic, we show that if R is a right (resp., left) perfect periodic ring and G is a locally finite group, then the group ring RG is periodic, too. We finally find some criteria under certain conditions when the tensor product of two periodic algebras over a commutative ring is again periodic. In addition, some other sorts of rings very close to periodic rings, namely the so-called weakly periodic rings, are also investigated. [ABSTRACT FROM AUTHOR]

In this paper, a new search technique based on a virus optimisation algorithm is proposed for calculating the neighbours of binary self-dual codes. The aim of this new technique is to calculate neighbours of self-dual codes without reducing the search field in the search process (this technique is known in the literature due to the computational time constraint) but still obtaining results in a reasonable time (significantly faster when compared to the standard linear computational search). We employ this new search algorithm to the well-known neighbour method and its extension, the $ k^{th} $-range neighbours, and search for binary $ [72, 36, 12] $ self-dual codes. In particular, we present six generator matrices of the form $ [I_{36} \ | \ \tau_6(v)], $ where $ I_{36} $ is the $ 36 \times 36 $ identity matrix, $ v $ is an element in the group matrix ring $ M_6(\mathbb{F}_2)G $ and $ G $ is a finite group of order 6, to which we employ the proposed algorithm and search for binary $ [72, 36, 12] $ self-dual codes directly over the finite field $ \mathbb{F}_2 $. We construct 1471 new Type I binary $ [72, 36, 12] $ self-dual codes with the rare parameters $ \gamma = 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 32 $ in their weight enumerators. [ABSTRACT FROM AUTHOR]

In this paper, we consider the ring of matrices $ \mathcal{A} $ of order $ 2 $ over the ring $ \mathbb{F}_2 [u] / \langle u^k \rangle $, where $ u $ is an indeterminate with $ u^k = 0 $, i.e. $ \mathcal{A} = M_2 (\mathbb{F}_2 [u] / \langle u^k \rangle) $. We derive the structure theorem for cyclic codes of odd length $ n $ over the ring $ \mathcal{A} $ with the help of isometry map from $ \mathcal{A} $ to $ \mathbb{F}_4 [u, v] / \langle u^k, v^2, u v - v u \rangle $, where $ v $ is an indeterminate satisfying $ v^2 = 0 $ and $ u v = v u $. We define a map $ \theta $ which takes the linear codes of odd length $ n $ over $ \mathcal{A} $ to linear codes of even length $ 2 k n $ over $ \mathbb{F}_4 $. We also define a weight on the ring $ \mathcal{A} $ which is an extension of the weight defined over the ring $ M_2 (\mathbb{F}_2) $. An example is also given as applications to construct the linear codes of odd length $ n $ over $ \mathcal{A} $. [ABSTRACT FROM AUTHOR]

We present a survey of the methods aimed at solving matrix linear bilateral equations (and, in particular, Sylvester-type equations) over different domains and at describing the structure of their solutions. Our main attention is focused on the extension and generalization of the results obtained by the authors earlier. On the basis of the standard form of polynomial matrices with respect to semiscalar equivalence, we develop a method aimed at solving Sylvester-type matrix polynomial equations. The structure of these solutions is investigated. We select solutions of bounded degrees and present the conditions for the uniqueness of these solutions. A method for constructing the solutions of Sylvester matrix equations over adequate rings is developed and criteria for the uniqueness of solutions of a certain type are obtained. We also establish the conditions for the existence of solution to the Sylvester-type matrix equation in rings of triangular and block-triangular matrices over the commutative domain of principal ideals. [ABSTRACT FROM AUTHOR]

We give a new technique for constructing self-dual codes based on a block matrix whose blocks arise from group rings and orthogonal matrices. The technique can be used to construct self-dual codes over finite commutative Frobenius rings of characteristic 2. We give and prove the necessary conditions needed for the technique to produce self-dual codes. We also establish the connection between self-dual codes generated by the new technique and units in group rings. Using the construction together with the building-up construction, we obtain new extremal binary self-dual codes of lengths 64, 66 and 68 and new best known binary self-dual codes of length 80. [ABSTRACT FROM AUTHOR]

The article explores research findings akin to Amitsur’s theorem, asserting that any derivation within a matrix ring can be expressed as the sum of an inner derivation and a hereditary derivation. In most results related to rings and semirings, Birkenmeier’s semicentral idempotents play a crucial role. This article is intended for PhD students, postdocs, and researchers.

AbstractLet R be a commutative ring with identity. For a positive integer [Formula: see text] let [Formula: see text] be the set of all n × n matrices over R and [Formula: see text] be the set of all non-zero matrices of [Formula: see text] The zero-divisor graph of [Formula: see text] is a simple directed graph with vertex set the non-zero zero-divisors in [Formula: see text] and two distinct matrices A and B are adjacent if their product is zero. Given a matrix [Formula: see text] Tr(A) is the trace of the matrix A. The trace graph of the matrix ring [Formula: see text] denoted by [Formula: see text] is the simple undirected graph with vertex set [Formula: see text] [Formula: see text] and two distinct vertices A and B are adjacent if and only if Tr [Formula: see text] For an ideal I of R, the notion of the ideal based trace graph, denoted by [Formula: see text] is a simple undirected graph with vertex set [Formula: see text] and two distinct vertices A and B are adjacent if and only if Tr [Formula: see text] In this survey, we present several results concerning the zero-divisor graph, trace graph and the ideal based trace graph of matrices over R.

Let R be a ring of characteristic different from 2 , m , n , s ≥ 1 fixed positive integers, L a noncentral Lie ideal of R and F : R → R a nonzero generalized skew derivation of R. We prove the following results: (a) If R is prime and there exists 0 ≠ a ∈ R such that a (F (x) m F (y) n − y n x m) s = 0 ∀ x , y ∈ L then R ⊆ M 2 (K) , the 2 × 2 matrix ring over a field K. (b) If R is semiprime and (F (x) m F (y) n − y n x m) s = 0 ∀ x , y ∈ L then either L centralizes a nonzero ideal of R or [ s 4 (x 1 , ... , x 4) , x 5 ] is a polynomial identity for R. [ABSTRACT FROM AUTHOR]

Recently, Anderson et al. in ([3], Comm. Alg. (2020)) introduced a generalization of multiplication modules called S-multiplication modules. In this article, the properties of S-prime submodules of S-multiplication modules are studied. Furthermore, we show that the concepts of simple modules, multiplication modules, and S-multiplication modules are equivalent in a torsion-free divisible module over an integral domain. Finally, we construct ring of matrices R' whose entries in a module M and demonstrate that M is S-multiplication R-module is equivalent to the existence of a specific ideal A of rings R' is S-multiplication ideal. [ABSTRACT FROM AUTHOR]

In 2020, Cossu and Zanardo raised a problem on the idempotent factorization of singular matrices in the form ( p z z ¯ ‖ z ‖ / p ) , where p is a prime integer which is irreducible but not prime in the ring of integers O K , with K a real quadratic number field, and z ∈ O K is such that 〈 p , z 〉 is a non-principal ideal. In this paper, we provide some classes of matrices that factor in a prescribed way and some other classes that do not. We further show that there are matrices in the above form that cannot be written as a product of two idempotent matrices. [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]

On the ring M n (C) of all n × n complex matrices A, the conjugate transpose involution A ↦ A * has the "properness" property A * A = 0 ⇒ A = 0 , as required for the existence of the partial order known as the " * -order" on M n (C) . More generally, related questions are asked and answered about the ring M n (R) of all n × n matrices over an arbitrary associative ring R with any given involution R → R (as a generalization of the conjugacy map C → C ), yielding a corresponding " * -transposition" involution on M n (R) . A criterion is found for this involution of M n (R) to be proper, so that M n (R) has a corresponding * -order. The special case R = Z k is also considered. [ABSTRACT FROM AUTHOR]

Given a finite dimensional algebra A over a field k , and a finite acyclic quiver Q , let Λ = A ⊗ k k Q / I , where kQ is the path algebra of Q over k and I is a monomial ideal. This paper is devoted to studying the separated monic representations over the bounded quiver (Q , I) over a subcategory X of A -mod, denoted by smon (Q , I , X). We show that (X , Y) is a (complete) hereditary cotorsion pair in A -mod if and only if (smon (Q , I , X) , rep (Q , I , Y)) is a (complete) hereditary cotorsion pair in Λ-mod. We also show that A is left weakly Gorenstein if and only if so is Λ. Provided that k Q / I is non-semisimple, the category ▪ of semi-Gorenstein-projective Λ-modules coincides with the category of separated monic representations ▪ if and only if A is left weakly Gorenstein. [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]

We define the generalized Galois inner product for codes over Frobenius rings, and then present the Galois linear complementary dual (LCD) codes over such rings which generalize the Galois LCD codes over finite fields defined in a recent reference. We describe the judging criterions for a generalized Galois LCD code over a Frobenius ring by introducing the rank of a matrix over such a ring. We also present necessary and sufficient conditions for the generalized Galois LCD codes over chain rings. The structure of the generalized Galois LCD codes over chain rings can be used to construct Galois LCD codes over finite fields. By using the structure of the generalized Galois LCD codes, we also give the description of constacyclic LCD codes over a class of chain rings. [ABSTRACT FROM AUTHOR]

In this paper, we study σ -skew McCoy rings under the σ -compatible or the σ -semicompatible conditions. We show that if R is a semicommutative right or left artinian ring which is σ -semicompatible with an epimorphism σ , then the Jacobson radical J (R) is σ -skew McCoy. As a corollary, we get that the Jacobson radical of a semicommutative artinian ring is right McCoy. We also show that every σ -compatible right duo ring is σ -skew McCoy and that for σ -compatible regular rings, the notions of the σ -skew McCoy and the right McCoy coincide. In addition, we show that every σ -semicompatible semicommutative ring is linearly σ -skew Camillo and that every matrix ring over a division ring is linearly σ -skew Camillo for any endomorphism σ. [ABSTRACT FROM AUTHOR]

The Drazin inverse is connected with the notion of index and core-nilpotent decomposition whenever it is discussed in the context of ring of matrices over complex field. In the absence of Drazin inverse for a given element from an arbitrary associative ring (not necessarily with unity), in this paper, the notion of right (left) core-nilpotent decomposition has been introduced and established its relations with right (left) π -regular property. In fact, the class of such decomposition has been characterized. In case of regular ring, observed that an element is right (left) π -regular if and only if it has a right (left) core-nilpotent decomposition. In the process, several properties of sharp order in an associative ring are studied and with the help of the same, new characterizations of Drazin inverse over an associative ring are obtained and the relation between core-nilpotent decomposition and the Drazin inverse is obtained. [ABSTRACT FROM AUTHOR]