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We give an explicit description of a diagonal map on the Bardzell resolution for any monomial algebra, and we use this diagonal map to describe the cup product on Hochschild cohomology. Then, we prove that the cup product is zero in positive degrees for triangular monomial algebras. Our proof uses the graded-commutativity of the cup product on Hochschild cohomology and does not rely on explicit computation of the Hochschild cohomology modules. [ABSTRACT FROM AUTHOR]

In the variety of not necessarily associative rings, we show that the sixty Bol-Moufang identities determine exactly thirteen subvarieties, and we determine all inclusions and necessary counterexamples. We make use of the concepts of magma rings and linear identities. The result is analogous to the classification of varieties of loops or quasigroups of Bol-Moufang type (Phillips, Vojtěchovský-2005) and was motivated by the study of tangent algebras to smooth loops of Bol-Moufang type (Mal'cev-1955, Sagle-1961, Nagy-1992, Bremner, Madariaga-2014). [ABSTRACT FROM AUTHOR]

We prove that cellular Noetherian algebras with finite global dimension are split quasi-hereditary over a regular commutative Noetherian ring with finite Krull dimension and their quasi-hereditary structure is unique, up to equivalence. In the process, we establish that a split quasi-hereditary algebra is semi-perfect if and only if the ground ring is a local commutative Noetherian ring. We give a formula to determine the global dimension of a split quasi-hereditary algebra over a commutative regular Noetherian ring (with finite Krull dimension) in terms of the ground ring and finite-dimensional split quasi-hereditary algebras. For the general case, we give upper bounds for the finitistic dimension of split quasi-hereditary algebras over arbitrary commutative Noetherian rings. We apply these results to Schur algebras over regular Noetherian rings and to Schur algebras over quotients rings of the integers. [ABSTRACT FROM AUTHOR]

We produce a fully faithful functor from finite type nilpotent spaces to cosimplicial binomial rings, thus giving an algebraic model of integral homotopy types. As an application, we construct an integral version of the Grothendieck–Teichmüller group. [ABSTRACT FROM AUTHOR]

In order to mine the key nodes in the spatial information network in the real task scenario and protect them, based on the mission tasks undertaken by spatial information network in the context of system, node types and information flow relationships are distinguished with the help of operational ring theory. Considering the high dynamic nature of spatial information network, time-varying graph analysis method is used to model the network. Based on the spatial information dynamic network model, the value factor of the operational ring is given by considering the formation time of the operational ring and combining with the traditional complex network indicators, the importance evaluation index of the spatial information network nodes is proposed as time-varying ring betweenness. The effectiveness and rationality of the proposed method are verified by comparing the node importance under indexes of time-varying ring betweenness, degree centrality and number of operational rings through experimental simulation. [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]

A method is proposed that reduces the computation of the reduced digital convolution operation to a set of independent convolutions computed in Galois fields. The reduced digital convolution is understood as a modified convolution operation whose result is a function taking discrete values in the same discrete scale as the original functions. The method is based on the use of partial convolutions, reduced to computing a modulo integer q 0 , which is the product of several prime numbers: q 0 = p 1 p 2 ... p n . It is shown that it is appropriate to use the expansion of the number q 0 , to q = p 0 p 1 p 2 ... p n , where p 0 is an additional prime number, to compute the reduced digital convolution. This corresponds to the use of additional digits in the number system used to convert to partial convolutions. The inverse procedure, i.e., reducing the result of calculations modulo q to the result corresponding to calculations modulo q 0 , is provided by the formula that used only integers proved in this paper. The possibilities of practical application of the obtained results are discussed. [ABSTRACT FROM AUTHOR]

We present a proof that all straight domains are locally divided—thereby answering two open problems posed by Dobbs and Picavet, which appeared in the survey "Open Problems in Commutative Ring Theory" written by Cahen, Fontana, Frisch, and Glaz. In fact, we are able to prove a stronger result: a prime ideal of a domain is straight if and only if it is locally divided. [ABSTRACT FROM AUTHOR]

In this article, we show that Mori domains, pseudo-valuation domains, and n-absorbing ideals, the three seemingly unrelated notions in commutative ring theory, are interconnected. In particular, we prove that an integral domain R is a Mori locally pseudo-valuation domain if and only if each proper ideal of R is a finite product of 2-absorbing ideals of R. Moreover, every ideal of a Mori locally almost pseudo-valuation domain can be written as a finite product of 3-absorbing ideals. To provide concrete examples of such rings, we study rings of the form A + XB [ X ] where A is a subring of a commutative ring B and X is indeterminate, which is of independent interest, and along with several characterization theorems, we prove that in such a ring, each proper ideal is a finite product of n-absorbing ideals for some n ≥ 2 if and only if A ⊆ B is essentially a finite product of field extensions. A complete description of when an order of a quadratic number field is a locally pseudo valuation domain, a locally almost pseudo valuation domain or a locally conducive domain is given. [ABSTRACT FROM AUTHOR]

This article explores the concept of the power serieswise Armendariz graph of a commutative ring, denoted as PA(R). It discusses various properties of PA(R), including diameter, clique, and girth, and examines the relationship between the ring-theoretic properties of R and the graph-theoretic properties of PA(R). The article provides examples and proofs to support its findings. It also discusses the relationship between PA(R) and the graph of the ring R[[X]], denoted as Γ(R[[X]]), and establishes several equivalences between different statements about the graph. The article concludes by discussing the implications of these equivalences and providing a list of references for further reading. [Extracted from the article]

Let R be a prime ring and Z(R) denotes the center of R. In this study, we expose the commutativity of R as a consequence of specific differential identities involving derivations acting on left ideals of R. Finally, we give examples that demonstrate the necessity of hypotheses taken in the theorems. [ABSTRACT FROM AUTHOR]

We classify nilpotent pre-Lie rings of cardinality p 4 and thereby braces of the same cardinality, for a sufficiently large prime p. It has been shown that nilpotent pre-Lie rings of cardinality p n correspond to strongly nilpotent braces of the same cardinality, for sufficiently large p. These braces are explicitly obtained from the corresponding pre-Lie rings by the construction of the group of flows. Braces of cardinality p 4 that are not right nilpotent have already been classified. Consequently, our results complete the classification of braces with such cardinality. [ABSTRACT FROM AUTHOR]

In this paper, we have defined the concept of minimal units in finite two-fold fuzzy neutrosophic rings modulo integers as a generalization of classical elements of the group of units of the mentioned neutrosophic rings. We apply our results to characterize all minimal units in the following two-fold neutrosophic rings (Zn (I)) fI for n ∈ {2,3,4,5}. [ABSTRACT FROM AUTHOR]

We introduce a new class of neutrosophic crisp set, and then it presented some operations via this sets like, NCT\intersection, NCT -union and algebraic NCT -difference to arrive at the algebraic ring construction. Also, we introduce the NCT - points, which showed that a NCT -set is NCT -union of its NCT -points, and so did we introduced the concept of the function on this sets, called NCT - function and some of their important properties. Finally, we introduced the concept of the topology and some of the concepts entrusted to it, as an introduction to those spaces that can be studied in detail in the future. [ABSTRACT FROM AUTHOR]

In this paper, we develop an explicit method to express finite algebraic numbers (in particular, certain idempotents among them) in terms of linear recurrent sequences, and give applications to the characterization of the splitting primes in a given finite Galois extension over the rational field. [ABSTRACT FROM AUTHOR]

We prove a purity property in telescopically localized algebraic K-theory of ring spectra: For n\geq 1, the T(n)-localization of K(R) only depends on the T(0)\oplus \dots \oplus T(n)-localization of R. This complements a classical result of Waldhausen in rational K-theory. Combining our result with work of Clausen–Mathew–Naumann–Noel, one finds that L_{T(n)}K(R) in fact only depends on the T(n-1)\oplus T(n)-localization of R, again for n \geq 1. As consequences, we deduce several vanishing results for telescopically localized K-theory, as well as an equivalence between K(R) and TC(\tau _{\geq 0} R) after T(n)-localization for n\geq 2. [ABSTRACT FROM AUTHOR]

It is well known that the ring radical theory can be approached via language of modules. In this work, we present some generalizations of classical results from module theory, in the two-sided and graded sense. Let G be a group, F an algebraically closed field with char (F) = 0 , A a finite dimensional G-graded associative F -algebra and M a G-graded unitary A -bimodule. We proved that if A = M n (F σ [ H ]) with an elementary-canonical G-grading, where H is a finite abelian subgroup of G and σ ∈ Z 2 (H , F *) , then M being irreducible graded implies that there exists a nonzero homogeneous element w ∈ M satisfying M = B w and B w = w B . Another result we proved generalizes the last one: if G is abelian, A is simple graded and M is finitely generated, then there exist nonzero homogeneous elements w 1 , w 2 , ... , w n ∈ M such that M = A w 1 ⊕ A w 2 ⊕ ⋯ ⊕ A w n , where w i A = A w i ≠ 0 for all i = 1 , 2 , ... , n , and each A w i is irreducible. The elements wi's are associated with the irreducible characters of G. We also describe graded bimodules over graded semisimple algebras. And we finish by presenting a Pierce decomposition of the graded Jacobson radical of any finite dimensional F -algebra with a G-grading. [ABSTRACT FROM AUTHOR]

By 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]

In this paper, we examine the notions of mj-clean ring and strongly mj-clean ring. And we will provide some of its basic properties. We examine the relationship of mj-clean ring with m-clean ring and j-clean ring. We prove that R is strongly mj-clean ring if and only if Mn(R) is strongly mj-clean ring. We prove that mj-clean ring is Dedekind-finite; i.e., ab = 1 implies that ba = 1. [ABSTRACT FROM AUTHOR]

Let m , n {m,n} be the fixed positive integers and let ℛ {\mathcal{R}} be a ring. In 1978, Herstein proved that a 2-torsion free prime ring ℛ {\mathcal{R}} is commutative if there is a nonzero derivation d of R such that [ d ⁢ ( ϱ ) , d ⁢ ( ξ ) ] = 0 {[d(\varrho),d(\xi)]=0} for all ϱ , ξ ∈ R {\varrho,\xi\in R} . In this article, we study the above mentioned classical result for higher derivations and describe the structure of semiprime rings by using the invariance property of prime ideals under higher derivations. Precisely, apart from proving some other important results, we prove the following. Let ( d i ) i ∈ ℕ {(d_{i})_{i\in\mathbb{N}}} and ( g j ) j ∈ ℕ {(g_{j})_{j\in\mathbb{N}}} be two higher derivations of semiprime ring ℛ {\mathcal{R}} such that [ d n ⁢ ( ϱ ) , g m ⁢ ( ξ ) ] ∈ Z ⁢ ( ℛ ) {[d_{n}(\varrho),g_{m}(\xi)]\in Z(\mathcal{R})} for all ϱ , ξ ∈ ℐ {\varrho,\xi\in\mathcal{I}} , where ℐ {\mathcal{I}} is an ideal of ℛ {\mathcal{R}} . Then either ℛ {\mathcal{R}} is commutative or some linear combination of ( d i ) i ∈ ℕ {(d_{i})_{i\in\mathbb{N}}} sends Z ⁢ ( ℛ ) {Z(\mathcal{R})} to zero or some linear combination of ( g j ) j ∈ ℕ {(g_{j})_{j\in\mathbb{N}}} sends Z ⁢ ( ℛ ) {Z(\mathcal{R})} to zero. We enrich our results with examples that show the necessity of their assumptions. Finally, we conclude our paper with a direction for further research. [ABSTRACT FROM AUTHOR]

Recently, the study of probabilities in ring theory has shown a significant increase in the field of algebra. Many interesting algebraic structures were modeled to find their probabilities in certain finite rings. In this paper, we introduce a new type of probability in finite rings, namely the squared-zero product probability. The aim is to study the square property and the zero product property of the ring. The focus of this study is the ring of matrices of dimension two over integers modulo prime p. To obtain the probability, the order of the square-annihilator of the ring is determined. The results found show that the squared-zero product probability of the ring is dependent on the value of p. [ABSTRACT FROM AUTHOR]

We introduce the notion of PC cancellative additive monoids with infinity and use it to characterize cancellative additive principal ideal domains with infinity. Our characterization improves various known characterizations from the literature, both, in the context of the commutative cancellative monoids, as well as in the context of the analogues of the statements from the commutative ring theory. [ABSTRACT FROM AUTHOR]

Modules are a generalization of the vector spaces of linear algebra over a ring instead of over a field of scalars. Some of the results in linear algebra are extremely useful in studying modules. In this paper, we highlight the use of Smith normal form in studying finitely generated modules over a univariate polynomial ring. As an application, we take advantage of Smith normal form to understand the structure of certain syzygy modules. [ABSTRACT FROM AUTHOR]

Let R be the commutative ring R = Zp²[x]/〈x²〉 with identity and Z*(R) be the set of all non-zero zero-divisors of R. Then, (R) is said to be a zero-divisor graph if and only if a · b = 0 where a; b 2 V (Γ(R)) = Z*(R) and (a; b) 2 E(Γ(R)). Let λ1, λ2,... λ2 be the eigenvalues of the adjacency matrix, and let µ1, µ2,..., µn be the eigenvalues of the Laplacian matrix of Γ(R). Then we discuss the energy E(Γ(R)) = Pn i=1jij and the Laplacian energy LE(Γ(R)) = Pn i=1 λi Γ 2m n where n and m are the order and size of Γ(R). [ABSTRACT FROM AUTHOR]

Let R be a (semi-) prime ring with extended centroid C, let f(X1, ... , Xk) be a multilinear polynomial over C in k noncommutative indeterminates which is not central-valued on R and let g be a generalized derivation of R. In this paper, we completely characterize the form of g and the structure of R such that (g(f(x1, ... , xk))m − γf(x1, ... , xk)n)s = 0 for all x1, ... , xk ∈ R, where γ ∈ C and m, n, s are fixed positive integers. Our results naturally improve and generalize the theorems obtained by Huang and Davvaz in [Generalized derivations of rings and Banach algebras, Comm. Algebra (2013); 43, 1188–1194] and the theorems recently obtained by De Filippis et al. in [Generalized derivations with nilpotent, power-central and invertible values in prime and semiprime rings, Comm. Algebra (2019); 47, 3025–3039]. Moreover, we describe a revised version of the theorem obtained by Huang in [On generalized derivations of prime and semiprime rings, Taiwanese J. Math. (2012); 16, 771–776.] [ABSTRACT FROM AUTHOR]

We point out that there is an error in the proof of Theorem 1.1 in [The discriminant of compositum of algebraic number fields, Int. J. Number Theory 15 (2019) 353–360]. We also prove that the result of this theorem holds with an additional hypothesis. However, it is an open problem whether the result of the theorem is true in general or not. [ABSTRACT FROM AUTHOR]

The combat domain of modern warfare is becoming increasingly multidimensional. It is important to evaluate the resilience of the air-ground cooperative network for defending against attack threats and recovery performance. First, a resilience analysis model was proposed to effectively analyze and evaluate the resilience of the air-ground cooperative network. Then, considering the available resources, three dynamic reconfiguration strategies were given from the global perspective to help the air-ground cooperative network quickly recover performance and enhance combat capabilities. Finally, a typical 50-node network was taken as an example to prove the effectiveness and feasibility of the proposed model. The proposed method can provide scientific guidance for improving the air-ground cooperative network combat capabilities. [ABSTRACT FROM AUTHOR]

In this paper, we propose a weak version of quotient for the algebraic action of a group on a variety, which we shall call a pseudo-quotient. They arise when we focus on the purely topological properties of good Geometric Invariant Theory (GIT) quotients regardless of their algebraic properties. The flexibility granted by their topological nature enables an easier identification in geometric constructions than classical GIT quotients. We obtain several results about the interplay between pseudo-quotients and good quotients. Additionally, we show that in characteristic zero pseudo-quotients are unique up to virtual class in the Grothendieck ring of algebraic varieties. As an application, we compute the virtual class of S L 2 (k) -character varieties for free groups and surface groups as well as their parabolic counterparts with punctures of Jordan type. [ABSTRACT FROM AUTHOR]

Let D be a Dedekind domain with finite residue fields. We provide topological insights into certain classes of ideals of Int(D) lying over a given maximal ideal \mathfrak m of D. We completely determine invertible/divisorial ideals in terms of topological properties of subsets of the \mathfrak m-adic completion of D. Moreover, these results are naturally extended to overrings of Int(D). As an application we provide explicit constructions of divisorial ideals of Int(D) which are not finitely generated. [ABSTRACT FROM AUTHOR]

Since the beginning of research on graph related to ring, many graphs have been constructed by the concept of rings to investigate the relation between graph theory and ring theory. The unity product graph associated with a ring R is a graph which is obtained by setting all unit elements of R as the vertex set and two distinct elements ri and rj are joined by an edge if and only if ri · rj = e. A subset C of the vertex set in a unity product graph is called a perfect code if for all ri 2 C, the unions of their closed neighbourhoods, S1(ri) yield the whole vertex set in such a way that any of two closed neighbourhoods have no common elements. In this paper, we determine different order perfect codes in unity product graphs associated with some commutative rings with identity. We classify the commutative rings with identity in which their associated unity product graphs admit the trivial (order 1) and non-trivial perfect codes. In addition, it is proved that the complement unity product graphs associated with the commutative rings with identity admit only the trivial perfect code. [ABSTRACT FROM AUTHOR]

The aim of this paper is to describe the topological K-ring in terms of generators and relations of a flag Bott manifold. We apply our results to give a presentation for the topological K-ring, and hence the Grothendieck ring of algebraic vector bundles over flag Bott–Samelson varieties. [ABSTRACT FROM AUTHOR]

Conway and Coxeter have shown that frieze patterns over positive rational integers are in bijection with triangulations of polygons. An investigation of frieze patterns over other subsets of the complex numbers has recently been initiated by Jørgensen and the first two authors. In this paper, we first show that a ring of algebraic numbers has finitely many units if and only if it is an order in a quadratic number field Q(d)$\mathbb {Q}(\sqrt {d})$ where d<0$d<0$. We conclude that these are exactly the rings of algebraic numbers over which there are finitely many non‐zero frieze patterns for any given height. We then show that apart from the cases d∈{−1,−2,−3,−7,−11}$d\in \lbrace -1,-2,-3,-7,-11\rbrace$ all non‐zero frieze patterns over the rings of integers Od$\mathcal {O}_d$ for d<0$d<0$ have only integral entries and hence are known as (twisted) Conway–Coxeter frieze patterns. [ABSTRACT FROM AUTHOR]

The paper introduces the left and right versions of the large class of Drazin inverses in terms of the left and right annihilators in a ring, which are called left-Drazin and right-Drazin inverses. We characterize some basic properties of these one-sided Drazin inverses, and discuss Jacobson's lemma for them. In addition, the relation between the Drazin inverses and these two one-sided inverses is given by means of the spectrum and the operator decomposition. As an application, the left-Drazin and right-Drazin invertibilities in the Calkin algebra are investigated. [ABSTRACT FROM AUTHOR]

We generalize Grothendieck's generic freeness lemma to modules over semi-constructible extensions of normally bounded domains, extending previous results of Irving, McConnell, Robson, Artin, Small and Zhang, in which the base ring is commutative. [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]

Let Δ be an (abstract) simplicial complex on n vertices. One can define the Artinian monomial algebra A (Δ) = k [ x 1 , ... , x n ] / 〈 x 1 2 , ... , x n 2 , I Δ 〉 , where k is a field of characteristic 0 and I Δ is the Stanley-Reisner ideal associated to Δ. In this paper, we aim to characterize the Weak Lefschetz Property (WLP) of A (Δ) in terms of the simplicial complex Δ. We are able to completely analyze when the WLP holds in degree 1, complementing work by Migliore, Nagel and Schenck on the WLP for quotients by quadratic monomials. We give a complete characterization of all 2-dimensional pseudomanifolds Δ such that A (Δ) satisfies the WLP. We also construct Artinian Gorenstein algebras that fail the WLP by combining our results and the standard technique of Nagata idealization. [ABSTRACT FROM AUTHOR]

A ring R is called right pseudo-injective if every R-monomorphism from a right ideal I of R to R R can be extended to an endomorphism of R R . In this short note, it is proved that if K is a field with characteristic 0, then the group algebra KG is right pseudo-injective if and only if G is finite. [ABSTRACT FROM AUTHOR]

The 2nd law of thermodynamics yields an irreversible increase in entropy until thermal equilibrium is achieved. This irreversible increase is often assumed to require large and complex systems to emerge from the reversible microscopic laws of physics. We test this assumption using simulations and theory of a 1D ring of N Ising spins coupled to an explicit heat bath of N Einstein oscillators. The simplicity of this system allows the exact entropy to be calculated for the spins and the heat bath for any N , with dynamics that is readily altered from reversible to irreversible. We find thermal-equilibrium behavior in the thermodynamic limit, and in systems as small as N = 2 , but both results require microscopic dynamics that is intrinsically irreversible. [ABSTRACT FROM AUTHOR]

A cone normed space is one of the generalizations of normed space in which the codomain of normed space is replaced by an ordered real Banach space. In this paper, some properties of the dual, in particular bi-dual in the setting of cone normed space were studied. This allows us to define the notion of reflexivity with respect to cone normed space and proved the necessary and sufficient condition for a cone normed space to be reflexive. [ABSTRACT FROM AUTHOR]

The representation theory of groups is one of the most interesting examples of the interaction between physics and pure mathematics, where group rings play the main role. The group ring RΓ is actually an associative ring that inherits the properties of the group Γ and the ring of coefficients R. In addition to the fact that the theory of group rings is clearly the meeting point of group theory and ring theory, it also has applications in algebraic topology, homological algebra, algebraic K-theory and algebraic coding theory. In this article, we provide a complete description of Gorenstein flat-cotorsion modules over the group ring RΓ, where Γ is a group and R is a commutative ring. It will be shown that if Γ' ≥ Γ is a finite-index subgroup, then the restriction of scalars along the ring homomorphism RΓ' - RΓ as well as its right adjoint RΓ RΓ RΓ', preserve the class of Gorenstein flat-cotorsion modules. Then, as a result, Serre's Theorem is proved for the invariant GhdRΓ, which refines the Gorenstein homological dimension of Γ over R, GhdRΓ, and is defined using flat-cotorsion modules. Moreover, we show that the inequality GFCD(RΓ) ≤ GFCD(R)+cdRΓ holds for the group ring RΓ, where GFCD(R) denotes the supremum of Gorenstein flat-cotorsion dimensions of all R-modules and cdRΓ is the cohomological dimension of Γ over R. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with nonzero identity. In this paper, we introduce and investigate a new class of ideals that is closely related to the class of r -ideals. A proper ideal I of R is said to be a (1 , r) -ideal if whenever nonunit elements a , b , c ∈ R and a b c ∈ I , then a b ∈ I or c ∈ Z (R). Some basic properties of (1 , r) -ideals are studied. For instance, we give a method to construct (1 , r) -ideals that are not r -ideals. Among other things, it is shown that if R admits a (1 , r) -ideal that is not an r -ideal, then R is a local ring. Also, we provide a necessary and sufficient condition in term of (1 , r) -ideals for a ring to be a total quotient ring and we determinate the (1 , r) -ideals of a chained ring. Finally, we give an idea about some (1 , r) -ideals of the localization of rings, the trivial ring extensions and the power series rings to construct nontrivial and original examples of (1 , r) -ideals. [ABSTRACT FROM AUTHOR]

Analogy has played an important role in developing modern mathematics. However, it is unclear to what extent students are granted opportunities to productively reason by analogy. This article proposes a set of lessons for introducing topics in ring theory that allow students to engage with the process of reasoning by analogy while exploring new (to the students) mathematics. In this way, students come to creatively establish new concepts that they may take ownership of. I provide insights from previous implementations and conclude by reflecting on what has (and has not) worked well in my experience with implementing the lesson. [ABSTRACT FROM AUTHOR]

This paper studies λ-constacyclic codes and skew λ-constacyclic codes over the finite commutative non-chain ring R = Z4 + uZ4 + u²Z4 with u3 = 0 for λ = (1 + 2u + 2u²) and (3 + 2u + 2u²). We introduce distinct Gray maps and show that the Gray images of λ-constacyclic codes are cyclic, quasi-cyclic, and permutation equivalent to quasi-cyclic codes over Z4. It is also shown that the Gray images of skew λ-constacyclic codes are quasi-cyclic codes of length 2n and index 2 over Z4. Moreover, the structure of λ-constacyclic codes of odd length n over the ring R is determined and give some suitable examples. [ABSTRACT FROM AUTHOR]

In recent years, soft sets have been widely used in many important decisionmaking real-life problems. In this paper, observing the usage of soft sets in such kind of vital problems, we have introduced the bijective-unitary bijective soft rings. Firstly, we have defined and exemplified a bijective soft ring and a unitary bijective soft ring. Moreover, we have presented some applications of bijective soft rings. We have shown the usage of bijective soft rings in coding theory. In this context, we have observed that by obtaining a bijective soft ring over a finite ring, we have a coding matrix to encode a given set of messages. Besides these applicable results, we have also obtained some relations between bijective soft and classical rings. [ABSTRACT FROM AUTHOR]

In this study, we prove the existence of the central Lie bi-derivation for the ring with identity on the quaternion ring. We also describe the triple Lie derivation using the Jordan derivation on the aforementioned ring. An example is provided to demonstrate that our result is theoretically viable. [ABSTRACT FROM AUTHOR]

In this paper we introduce two new classes of inverses for square matrices, which are called 1Drazin (in short, 1D) inverse and Drazin1 (in short, D1) inverse. Next, we investigated the existence and uniqueness of 1D inverse and its dual D1 inverse. Some representation and characterizations of these inverses are derived. In addition to this, we obtain some properties of 1D and D1 inverses through idempotent and binary relations. [ABSTRACT FROM AUTHOR]