Narrow Results

In this paper, we define several versions of the left extending condition for nearrings and rings not necessarily having a unity. We investigate the interconnections between the extending conditions and the Baer annihilating conditions in a right nearring. Our results culminate in a nearring version of the Chatters–Khuri theorem for rings. Examples are provided illustrating and delimiting our results.

The nature of universe problems is pretty ambiguous in almost all disciplines. To overcome the uncertainties, fuzzy set theory was developed. Later, mathematicians developed new concept, hybrid structures. The main aim of this paper is to study about hybrid structures in rings. The ideas of hybrid ideals, hybrid duo and hybrid prime ideals of rings are presented in this paper and investigate its properties. We found an equivalent claims for a ring to be regular and intra-regular. Further, we prove that hybrid product of hybrid right ideal and hybrid left ideal will coincide with hybrid intersection of hybrid right ideal and hybrid left ideal under certain condition. Moreover, we examine results related to hybrid prime ideals in rings.

It is well known that a solution for the Firing Squad Synchronization Problem (FSSP) takes time 2n - 1, n being the number of soldiers. It is also known that a solution on a square of n × n soldiers takes the same time. The main result of this paper is a new solution for the FSSP on networks shaped as squares. Our solution is optimal in two aspects: it is communication optimal (the so-called 1-bit solution) and is also time optimal, as it takes 2n - 1 time. It is also used as a building block to construct a very efficient solution on the square torus. Our approach applies also to linearly shaped networks and yields on the ring almost optimal time & communication solutions. In particular, for the square torus with n rows and rings of n processors, if n is even our solution is time & communication optimal. Otherwise, it is communication-optimal but does not match the lower bound on the time of a synchronization just by 1 time unit.

Relativistic thick ring models are constructed using previously found analytical Newtonian potential-density pairs for flat rings and toroidal structures obtained from Kuzmin–Toomre family of disks. In particular, we present systems with one ring, two rings and a disk with a ring. Also, the circular velocity of a test particle and its stability when performing circular orbits are presented in all these models. In general, we find that regions of non-stability appear between the rings when they become thinner.

The Event Horizon Telescope’s image of the M87 black hole provides an exciting opportunity to study black hole physics. Since a black hole’s event horizon absorbs all electromagnetic waves, it is difficult to actively probe the horizon’s existence. However, with the help of a family of extremely compact, horizon-less objects, named “gravastars”, whose external spacetimes are nearly identical to those of black holes, one can test the absence of event horizons: absences of additional features that arise due to the existence of the gravastar, or its surface, can be used as quantitative evidence for black holes. We apply Gralla et al. approach of studying black hole images to study the images of two types of gravastars: transparent ones and reflective ones. In both cases, the transmission of rays through gravastars, or their reflections on their surfaces, leads to more rings in their images. For simple emission models, where the redshifted emissivity of the disk is peaked at a particular radius $r_{\mathrm{\text{peak}}}$, the position of a series of rings can be related in a simple manner to light ray propagation: a ring shows up around impact parameter $b$ whenever rays incident from infinity at $b$ intersects the disk at $r_{\mathrm{\text{peak}}}$. We show that additional rings will appear in the images of transparent and reflective gravastars. In particular, one of the additional rings for the reflective gravastar is due to the prompt reflection of light on the gravastar surface, and appears to be well separated from the others. This can be an intuitive feature, which may be reliably used to constrain the reflectivity of the black hole’s horizon.

We characterize rings over which every cotorsion module is pure injective (Xu rings) in terms of certain descending chain conditions and the Ziegler spectrum, which renders the classes of von Neumann regular rings and of pure semisimple rings as two possible extremes. As preparation, descriptions of pure projective and Mittag–Leffler preenvelopes with respect to so-called definable subcategories and of pure generation for such are derived, which may be of interest on their own. Infinitary axiomatizations lead to coherence results previously known for the special case of flat modules. Along with pseudoflat modules we introduce quasiflat modules, which arise naturally in the model-theoretic and the category-theoretic contexts.

A derivation of an associative ring R is an additive map satisfying T(xy) = T(x)y + xT(y) for all x, y in R. We study rings with a derivation T satisfying Herstein's condition [T(R), T(R)] = 0. (The commutator [u, v] is defined by: [u, v] = uv - vu.) This work studies the structure of the ideal I generated by T(R). We show that I³ is in the center of R, and we show that R has an ideal K which is contained in the kernel of T, K² = 0, and [T(R/K), R/K] generates a trivial ideal of R/K.

Let $S$ be a semigroup and $b,c\in S$. The concept of $(b,c)$-inverses was introduced by Drazin in 2012. It is well known that the Moore–Penrose inverse, the Drazin inverse, the Bott–Duffin inverse, the inverse along an element, the core inverse and dual core inverse are all special cases of the $(b,c)$-inverse. In this paper, a new relationship between the $(b,c)$-inverse and the Bott–Duffin $(e,f)$-inverse is established. The relations between the $(b,c)$-inverse of $paq$ and certain classes of generalized inverses of $pa$ and $aq$, and the $(b^{\prime },c^{\prime })$-inverse of $a$ are characterized for some $b^{\prime },c^{\prime }\in S$, where $p,a,q\in S$. Necessary and sufficient conditions for the existence of the $(B,C)$-inverse of a lower triangular matrix over an associative ring $R$ are also given, and its expression is derived, where $B,C$ are regular triangular matrices.

Given a complete modular meet-continuous lattice $A$, an inflator on $A$ is a monotone function $d:A\to A$ such that $a\le d(a)$ for all $a\in A$. If $I(A)$ is the set of all inflators on $A$, then $I(A)$ is a complete lattice. Motivated by preradical theory, we introduce two operators, the totalizer and the equalizer. We obtain some properties of these operators and see how they are related to the structure of the lattice $A$ and with the concept of dimension.

We classify modules and rings with some specific properties of their intersection graphs. In particular, we describe rings with infinite intersection graphs containing maximal left ideals of finite degree. This answers a question raised in [S. Akbari, R. Nikandish and J. Nikmehr, Some results on the intersection graphs of ideals of rings, J. Algebra Appl.12 (2013) 1250200]. We also generalize this result to modules, i.e. we get the structure theorem of modules for which their intersection graphs are infinite and contain maximal submodules of finite degree. Furthermore, we omit the assumption of maximality of submodules and still get a satisfactory characterization of such modules. In addition, we show that if the intersection graph of a module is infinite but its clique number is finite, then the clique and chromatic numbers of the graph coincide. This fact was known earlier only in some particular cases. It appears that such equality holds also in the complement graph.

Here, we continue to characterize a recently introduced notion, le-modules ${}_{R}M$ over a commutative ring $R$ with unity [A. K. Bhuniya and M. Kumbhakar, Uniqueness of primary decompositions in Laskerian le-modules, Acta Math. Hunga.158(1) (2019) 202–215]. This paper introduces and characterizes Zariski topology on the set Spec$(M)$ of all prime submodule elements of $M$. Thus, we extend many results on Zariski topology for modules over a ring to le-modules. The topological space Spec$(M)$ is connected if and only if R/Ann(M) contains no idempotents other than $\overline{0}$ and $\overline{1}$. Open sets in the Zariski topology for the quotient ring R/Ann(M) induces a base of quasi-compact open sets for the Zariski topology on Spec$(M)$. Every irreducible closed subset of Spec$(M)$ has a generic point. Besides, we prove a number of different equivalent characterizations for Spec$(M)$ to be spectral.

There is a special local ring $E$ of order $4,$ without identity for the multiplication, defined by $E=〈a,b|2a=2b=0,a^2=a,b^2=b,ab=a,ba=b〉.$ We study the algebraic structure of linear codes over that non-commutative local ring, in particular their residue and torsion codes. We introduce the notion of quasi self-dual codes over $E,$ and Type IV codes, that is quasi self-dual codes whose all codewords have even Hamming weight. We study the weight enumerators of these codes by means of invariant theory, and classify them in short lengths.

Let $R$ be a ring and let $n$ be an arbitrary but fixed positive integer. We characterize those rings $R$ whose elements $a$ satisfy at least one of the relations that $a^n+a$ or $a^n-a$ is a nilpotent whenever $n\in \mathbb{N}\setminus \{1\}$. This extends results from the same branch obtained by Danchev [A characterization of weakly J(n)-rings, J. Math. Appl. 41 (2018) 53–61], Koşan et al. [Rings with $x^n-x$ nilpotent, J. Algebra Appl. 19 (2020)] and Abyzov and Tapkin [On rings with $x^n-x$ nilpotent, J. Algebra Appl. 21 (2022)], respectively.

Let $\mathcal{R}$ be a ∗-ring with the center 𝒵($\mathcal{R}$) and ℕ be the set of nonnegative integers. In this paper, it is shown that if $\mathcal{R}$ contains a nontrivial self-adjoint idempotent which admits a generalized ∗-Lie higher derivable mapping Δ = {G_n}_n∈ℕ associated with a ∗-Lie higher derivable mapping ℒ = {L_n}_n∈ℕ, then for any X, Y in $\mathcal{R}$ and for each n in ℕ there exists an element Z_X,Y (depending on X and Y) in the center 𝒵($\mathcal{R}$) such that G_n(X + Y ) = G_n(X) + G_n(Y) + Z_X,Y.

We give an explicit finite presentation of a group normally generated by SL_∞(ℤ). As a consequence, such a group cannot act on e.g. a finite dimensional contractible manifold or on a compact manifold.

This paper explains the important conceptual and technical differences between the method of p-cycles and two other recent advances involving a cyclic orientation to protection. These are enhanced rings and cycle double covers. The most fundamental difference that is unique to p-cycles is the aspect of straddling span failure protection. This enables mesh-like efficiency levels at well under 100% redundancy. In contrast enhanced rings and advanced cycle cover methods are both seeking to reduce span overlaps in what is otherwise a purely ring-like logical paradigm in which 100% redundancy remains the best that can possibly be achieved.

In this paper, we will review the developing features of computations based on rings. Particularly, we will analyse what kinds of interaction occur between gliders travelling on a ‘cyclotron’ cellular automaton derived from a catalog of collisions. We will demonstrate that collisions between gliders emulate the basic types of interaction that occur between localizations in non-linear media: fusion, elastic collision, and soliton-like collision. Computational outcomes of a swarm of gliders circling on a one-dimensional torus are analysed via implementation of some simple computing models. Gliders in one-dimensional cellular automata are compact groups of non-quiescent patterns translating along an automaton lattice. They are cellular-automaton analogous to localizations or quasi-local collective excitations travelling in a spatially extended non-linear medium. So, they can be represented as binary strings or symbols travelling along a one-dimensional ring, interacting with each other and changing their states, or symbolic values, as a result of interactions. We present a number of complex one-dimensional cellular automata with such features.