Narrow Results

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.

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.

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.

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.

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.

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.

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.

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.