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This article concerns commutative factor rings for ideals contained in the center. A ring $R$ is called CIFC if $R/I$ is commutative for some proper ideal $I$ of $R$ with $I\subseteq Z(R)$, where $Z(R)$ is the center of $R$. We prove that (i) for a CIFC ring $R$, $W(R)$ contains all nilpotent elements in $R$ (hence Köthe's conjecture holds for $R$) and $R/W(R)$ is a commutative reduced ring; (ii) $R$ is strongly bounded if $R/N_*(R)$ is commutative and $0\neq N_*(R)\subseteq Z(R)$, where $W(R)$ (resp., $N_*(R)$) is the Wedderburn (resp., prime) radical of $R$. We provide plenty of interesting examples that answer the questions raised in relation to the condition that $R/I$ is commutative and $I\subseteq Z(R)$. In addition, we study the structure of rings whose factor rings modulo nonzero proper ideals are commutative; such rings are called FC. We prove that if a non-prime FC ring is noncommutative then it is subdirectly irreducible.

Our question is what ring R which all modules over R are determined, up to isomorphism, by their endomorphism rings? Examples of this ring are division ring and simple Artinian ring. Any semi simple ring does not satisfy this property. We construct a semi simple ring R but R is not a simple Artinian ring which all modules over R are determined, up to isomorphism, by their endomorphism rings.

Let R be a nil algebra over a field of characteristic 0, and let δ be a derivation of R. Then the differential polynomial ring R[X,δ] cannot be mapped onto a unital simple ring homomorphically.

The Laplacian controllability of a family of graphs that are non-simple is studied in the paper. Without the regular assumption that the adjacency matrix is binary, the authors consider more flexible weighting parameters to represent the practical connection strength between nodes. Suppose the node states of the graphs evolve according to the Laplacian dynamics. The Laplacian eigenspaces of a class of ring graphs are explored, by which a sufficient condition to render the graphs controllable with the minimum number of input is proposed. Numerical examples are provided to illustrate the theoretical results.

Suppose that $R$ is an associative unital ring and that $E=(E^0,E^1,r,s)$ is a directed graph. Utilizing results from graded ring theory we show, that the associated Leavitt path algebra $L_R(E)$ is simple if and only if $R$ is simple, $E^0$ has no nontrivial hereditary and saturated subset, and every cycle in $E$ has an exit. We also give a complete description of the center of a simple Leavitt path algebra., Comment: 10 pages. From v1 to v2: Added references and updated Section 1

Let Q be a (not necessarily unital) simple ring or algebra. A nonempty subset S of Q is said to have zero product if S 2 = 0 . We classify all maximal zero product subsets of Q by proving that the map R ↦ R ∩ LeftAnn ( R ) is a bijection from the set of all proper nonzero annihilator right ideals of Q onto the set of all maximal zero product subsets of Q. We also describe the relationship between the maximal zero product subsets of Q and the maximal inner ideals of its associated Lie algebra.

A coupled two-ring-network with time delay is developed to avoid the weakness of a simple ring topology. The divide and conquer algorithm is used to analyze the distribution of the eigenvalues of the adjacent matrix. The stability of the equilibrium and inter-layer synchronization of the periodic dynamics are studied, and the dynamical interactions between coupled rings which lead to complicated mirror and reflection waves are further analyzed. Special attention is paid on the generalized synchronization between two rings when the adjacent matrix has repeated eigenvalues. Numerical simulations verify the theoretical results, and show that the appropriate settings of the time delay and transverse coupling can guarantee the “backup and recovery” procedure.

For a subset [Formula: see text] of a ring [Formula: see text] we denote by [Formula: see text] the ideal of [Formula: see text] generated by [Formula: see text]. Given a higher commutator [Formula: see text] of [Formula: see text], if [Formula: see text] then [Formula: see text]? The question is motivated by the result that a ring [Formula: see text] is equal to its subring generated by [Formula: see text] if [Formula: see text] is either a noncommutative simple ring (by Herstein) or a unital ring with [Formula: see text] (by Eroǧlu). In this note, we study the question for the rings [Formula: see text] satisfying the property that every proper ideal of [Formula: see text] is contained in a maximal ideal (in particular, if [Formula: see text] is finitely generated as an ideal).

If $M$ is a simple module over a ring $R$ then, by the Schur's lemma, the endomorphism ring of $M$ is a division ring. However, the converse of this result does not hold in general, even when $R$ is artinian. In this short note, we consider perfect rings for which the converse assertion is true, and we show that these rings are exactly the primary decomposable ones.

Angle measurement means are one of the advanced directions for the application of gas ring lasers. The requirements of ring lasers used in angle measurement devices are different, in many respects, from those used in navigation. A simple ring laser developed for implementation in high-precision angle measurement instruments is described. The main specifications are presented. Theoretical and experimental studies and computer simulation of ring laser parameters are presented. The studies were carried out, taking into account changes in the Earth’s rotation speed around the axis and without taking into account changes in the Earth’s rotation speed. The obtained results indicate that the change in the speed of rotation of the Earth affects the accuracy of a ring laser. In order to reduce this error, it is necessary to fulfill the requirements of the rotational axis of the ring laser relative to the rotational axis of the Earth. Methods of the accelerated tests are imperfect for determining a term of storage. Therefore, tests during a real storage term remain the most reliable. The operation of the ring lasers in angle measuring instruments for more than 20 years has demonstrated their high stability. Using the ring laser, the angle measurement means of accuracy that exceeds the accuracy of the existing National Standards for a plane angle can be developed.

We study networks of coupled oscillators and analyze the role of coupling delays in determining the emergence of cluster synchronization. Given a network topology and a particular arrangement of the coupling delays over the network connections, different patterns of cluster synchronization may emerge. We focus on a simple ring network of six bidirectionally coupled identical oscillators, for which with two different values of the delays, a total of eight cluster synchronization patterns may emerge, depending on the assignment of the delays to the ring connections. We analyze stability of each of the patterns and find that for large enough coupling strength and specific values of the delays, they can all be stabilized. We construct an experimental ring of six bidirectionally coupled Colpitts oscillators, with delayed connections obtained by coupling the oscillators via RF cables of appropriate length. We find that experimental observations of cluster synchronization are in essential agreement with theoretical predictions. We also verify our theory in a fully connected network of fifty nodes for which connections are randomly assigned to be either undelayed or delayed with a given probability.

International audience; We developed a type of toroidal multi-pass cell with multi-layer patterns based on the off-axis model. The effective path length of the original toroidal multi-pass cell is extended several roundtrips in comparison with the single-layer pattern, since the inner surface of the toroidal multi-pass cell is more efficiently utilized. The light pattern has been achieved by using the simple ring surface, which is easy to fabricate. The exact analytical equations for the design of the toroidal multi-pass cell were derived based on analytical vector calculations. A series of numerical ray tracing simulations is presented, and the maximum theoretical optical path length that can be reached is 30 m with a setup of 5 cm column radius. Furthermore, two practical spot patterns are demonstrated with a path length of 8.3 m for a two-layer pattern and 10 m for a three-layer pattern, with respective effective volumes of 63 mL and 94 mL. Furthermore, the fringe effect is substantially reduced to less than 0.5% by the usage of our designed mask.

In this paper we prove that which of Green's relations $\mathcal{L,R,H}$ and $\mathcal{D}$ in rings preserve the minimality of quasi-ideal. By this it is possible to show the structure of the classes generated by the above relations which have a minimal quasi ideal. For the completely simple rings we show that they are generated by the union of zero with a $\mathcal{D} $-class. Also we emphasize that a completely simple ring coincides with the union of zero with a $\mathcal{D} $-class if and only if it is a division ring.

An expanded porphyrinoid has been obtained by a simple ring expansion from a contracted porphyrinoid, namely corrole. Spectroscopic, structural, and computational investigations reveal peculiar π-conjugation and geometry. The effect of extended π-conjugation is evident from perturbed redox behavior and photophysical properties. Owing to the strong diatropic ring current of the corrole and cross-conjugation, the molecule exhibits a non-aromatic nature for the expanded π-circuit, as evident from NMR studies.

Let R be a unital simple ring. Under some technical restrictions, we characterize m-linear mappings G:Rm→R satisfying [G(u,…,u),u] = 0 for all unit u∈R.

An operationally simple ring closure of racemic 1,1′-bi-2-naphthol (BINOL) yielded the heterocyclic aromatic compound 7H-dibenzo[c,g]carbazole (DBC). This one-pot method gave a good conversion and is suitable for gram-scale synthesis. DBC derivatives have high thermal durability, amorphous and crystalline structures with unique morphological properties, and semi-conducting behavior with potential applications in organic electronics.

In this paper, we consider the ideal based zero divisor graph ΓI(R) of a commutative ring R. We discuss some graph theoretical properties of ΓI(R) in relation with zero divisor graph. We also relate certain parameters like vertex chromatic number, maximum degree and minimum degree for the graph ΓI(R) with that of Γ(R/I ). Further we determine a necessary and sufficient condition for the graph to be Eulerian and regular.

We prove that the ring of polynomials in several commuting indeterminates over a nil ring cannot be homomorphically mapped onto a ring with identity, i.e. it is Brown-McCoy radical. It answers a question posed by Puczylowski and Smoktunowicz. We also show that the central closure of a prime nil ring cannot be a simple ring with identity, solving a problem due to Beidar.

The Nagata ring R(X) and the Serre’s conjecture ring R⟨X⟩ are two localizations of the polynomial ring R[X] at the polynomials of unit content and at the monic polynomials, respectively. In this pa...

The purpose of the paper is to study prime rings R such that the central closure RC is a simple ring with 1 and it is finitely generated over R by elements of the extended centroid C, that is, R C = R [ c 1 , … , c n ] , for some c 1 , … , c n ∈ C . In particular, we will show that if there exists a prime ring with zero center whose central closure is simple with 1 and generated by finitely many central elements, then there exists such a ring whose central closure is generated by two central elements.

A nonassociative ring which contains a well-known associative ring or left symmetric ring also known as Vinberg ring is of great interest. A method to construct Vinberg nonassociative ring is given; Vinberg nonassociative ring is shown as simple; all the derivations of nonassociative simple Vinberg algebra defined are determined; and finally in solid algebra it is shown that if is a nonzero endomorphism of , then is an epimorphism.

The Gelfand–Kirillov dimension has gained importance since its introduction as a tool in the study of non-commutative infinite dimensional algebras and their modules. In this paper we show a dichotomy for the Gelfand–Kirillov dimension of simple modules over certain simple rings of differential operators. We thus answer a question of J. C. McConnell in Representations of solvable Lie algebras V. On the Gelfand-Kirillov dimension of simple modules. McConnell (J. Algebra 76(2), 489–493, 1982) concerning this dimension for a class of algebras that arise as simple homomorphic images of solvable lie algebras. We also determine the Gelfand–Kirillov dimension of an induced module.

In this paper, we introduce the ring of Colombeau full generalized quaternion, and we study its algebraic and topological properties. We prove that the ring of Colombeau full generalized quaternion...