Nilpotent Elements Research Articles

A Remark on Structure of Projective Klingenberg Spaces over a Certain Local Algebra

This article is devoted to the projective Klingenberg spaces over a local ring, which is a linear algebra generated by one nilpotent element. In this case, subspaces of such Klingenberg spaces are described. The notion of the “degree of neighborhood” is introduced. Using this, we present the geometric description of subsets of points of a projective Klingenberg space whose arithmetical representatives need not belong to a free submodule.

A Jordan canonical form for nilpotent elements in an arbitrary ring

In this paper we give an inductive new proof of the Jordan canonical form of a nilpotent element in an arbitrary ring. If a∈R is a nilpotent element of index n with von Neumann regular an−1, we decompose a=ea+(1−e)a with ea∈eRe≅Mn(S) a Jordan block of size n over a corner S of R, and (1−e)a nilpotent of index <n for an idempotent e of R commuting with a. This result makes it possible to characterize prime rings of bounded index n with a nilpotent element a∈R of index n and von Neumann regular an−1 as a matrix ring over a unital domain.

Nil-clean elements which are not clean in certain subrings of M3(ℤ)

Let R be a ring with identity. An element x ∈ R is said to be clean if x = u + e for some unit u and idempotent e in R. If x = u + e or x = u − e for some unit u and idempotent e in R, then x is said to be weakly clean. The element x is said to be nil-clean if it is the sum of an idempotent and a nilpotent element. The ring R is called clean (weakly clean, nil-clean) if all of its elements are clean (weakly clean, nil-clean, respectively). It is known that nil-clean rings are clean. However, a nil-clean element is not necessarily clean. We give more examples of this by showing the existence of elements which are nil-clean but not clean in a subring of M3(ℤ), the ring of 3 × 3 matrices over ℤ. As a consequence, we also obtain examples of weakly clean elements which are not clean.

Weak nil clean ideals

As a generalization of a nil clean ideal, we define a weak nil clean ideal of a ring. An ideal I of a ring R is said to be weak nil clean ideal if for any $$x\in I$$, either $$x=e+n$$ or $$x=-e+n$$, where n is a nilpotent element and e is an idempotent element of R. Some interesting properties of weak nil clean ideal and its relation with weak nil clean ring have been discussed.

Iterants and the Dirac Equation

This paper introduces the concept of iterant algebra and applies it to the formation of basic Clifford algebras including a reconstruction of the complex numbers in terms of a formalization of temporal process. Iterant algebra is shown to include all of matrix algebra and applications are given to a representation of the su(3) Lie algebra for the Standard Model and to the construction of the Dirac Equation. This construction of the Dirac Equation makes it clear how solutions arise from nilpotent elements in the Clifford algebra and how Fermion algebra and the algebra of Majorana Fermions emerges in this context. The paper ends with a formulation of the original Majorana Dirac Equation in terms of Clifford algebra in the context of iterants.

Associative rings in which nilpotents form an ideal

AbstractIt is shown that ifN(R) is a Lie ideal ofR(respectively Jordan ideal andRis 2-torsion-free), thenN(R) is an ideal. Also, it is presented a characterization of Noetherian NR rings with central idempotents (respectively with the commutative set of nilpotent elements, the Abelian unit group, the commutative commutator set).

On quasi-radical of near-ring of polynomials

Abstract The intersection of all maximal right ideals of a near-ring N is called the quasi-radical of N. In this paper, first we show that the quasi-radical of the zero-symmetric near-ring of polynomials R0[x] equals to the set of all nilpotent elements of R0[x], when R is a commutative ring with Nil (R)2 = 0. Then we show that the quasi-radical of R0[x] is a subset of the intersection of all maximal left ideals of R0[x]. Also, we give an example to show that for some commutative ring R the quasi-radical of R0[x] coincides with the intersection of all maximal left ideals of R0[x]. Moreover, we prove that the quasi-radical of R0[x] is the greatest quasi-regular (right) ideal of it.

Nilpotent networks and 4D RG flows

Starting from a general mathcal{N}=2 SCFT, we study the network of mathcal{N}=1 SCFTs obtained from relevant deformations by nilpotent mass parameters. We also study the case of flipper field deformations where the mass parameters are promoted to a chiral superfield, with nilpotent vev. Nilpotent elements of semi-simple algebras admit a partial ordering connected by a corresponding directed graph. We find strong evidence that the resulting fixed points are connected by a similar network of 4D RG flows. To illustrate these general concepts, we also present a full list of nilpotent deformations in the case of explicit mathcal{N}=2 SCFTs, including the case of a single D3-brane probing a D- or E-type F-theory 7-brane, and 6D (G, G) conformal matter compactified on a T2, as described by a single M5-brane probing a D- or E-type singularity. We also observe a number of numerical coincidences of independent interest, including a collection of theories with rational values for their conformal anomalies, as well as a surprisingly nearly constant value for the ratio aIR/cIR for the entire network of flows associated with a given UV mathcal{N}=2 SCFT. The arXiv submission also includes the full dataset of theories which can be accessed with a companion Mathematica script.

Hyperrings which every element is sum of an idempotent and nilpotent

In this paper, we define a generalized of clean of Krasner hyperrings for general hyperrings based on the notation of nilpotent elements of a general hyperring $R$, named nil clean hyperring. We examine characterization of this kind of hyperrings and finally, we obtain some relations of nil clean general hyperrings with other hyperrings.

A Characterization of Left Regularity

We show that a zero-symmetric near-ring $N$ is left regular if and only if $N $ is regular and isomorphic to a subdirect product of integral near-rings, where each component is either an Anshel-Clay near-ring or a trivial integral near-ring. We also show that a zero-symmetric near-ring is regular without nonzero nilpotent elements if and only if the multiplicative semigroup of N is a union of disjoint groups.

Holographic duals of 6d RG flows

A notable class of superconformal theories (SCFTs) in six dimensions is parameterized by an integer N , an ADE group G, and two nilpotent elements μL,R in G. Nilpotent elements have a natural partial ordering, which has been conjectured to coincide with the hierarchy of renormalization-group flows among the SCFTs. In this paper we test this conjecture for G = SU(k), where AdS7 duals exist in IIA. We work with a seven-dimensional gauged supergravity, consisting of the gravity multiplet and two SU(k) non-Abelian vector multiplets. We show that this theory has many supersymmetric AdS7 vacua, determined by two nilpotent elements, which are naturally interpreted as IIA AdS7 solutions. The BPS equations for domain walls connecting two such vacua can be solved analytically, up to a Nahm equation with certain boundary conditions. The latter admit a solution connecting two vacua if and only if the corresponding nilpotent elements are related by the natural partial ordering, in agreement with the field theory conjecture.

Finite generation of Lie derived powers of associative algebras

Let [Formula: see text] be an associative algebra over a field of characteristic [Formula: see text] that is generated by a finite collection of nilpotent elements. We prove that all Lie derived powers of [Formula: see text] are finitely generated Lie algebras.

Minimal $W$-Superalgebras and the Modular Representations of Basic Lie Superalgebras

Let $\mathfrak g = \mathfrak g\_\bar 0 + \mathfrak g\_\bar 1$ be a basic Lie superalgebra over $\mathbb{C}$, and $e$ a minimal nilpotent element in $\mathfrak g\_\bar 0$. Set $W\_\chi'$ to be the refined $W$-superalgebra associated with the pair $\mathfrak g,e)$, which is called a minimal $W$-superalgebra. In this paper we present a set of explicit generators of minimal $W$-superalgebras and the commutators between them. In virtue of this, we show that over an algebraically closed field $\mathbb k$ of characteristic $p \gg 0$, the lower bounds of dimensions in the modular representations of basic Lie superalgebras with minimal nilpotent $p$-characters are attainable. Such lower bounds are indicated in \[33] as the super Kac–Weisfeiler property.

NC-Rings and Some Commutativity Conditions

Sum of two nilpotent elements in a ring may not be nilpotent in general, but for commutative rings this sum is nilpotent. In between commutative and non-commutative rings there are several types of rings in which this property holds. For instance, reduced, NI, AI (or IFP), 2-primal, reversible and symmetric, etc. We may term these types of rings as nearby commutative rings (in short NC-rings). In this work we have studied properties and various characterizations of such rings as well as rngs. As applications, we have investigated some commutativity conditions by involving semi-projective-Morita-contexts and right Ck-Goldie rings.

A note on the FIP property for extensions of commutative rings

A ring extension R ? S is said to be FIP if it has only finitely many intermediate rings between R and S. The main purpose of this paper is to characterize the FIP property for a ring extension, where R is not (necessarily) an integral domain and S may not be an integral domain. Precisely, we establish a generalization of the classical Primitive Element Theorem for an arbitrary ring extension. Also, various sufficient and necessary conditions are given for a ring extension to have or not to have FIP, where S = R[?] with ? a nilpotent element of S.

Reflexive-nilpotents-property skewed by ring endomorphisms

The reflexive property for rings was introduced by Mason and play roles in noncommutative ring theory. A ring R is called reflexive if for a, b in R, aRb = 0 implies bRa = 0. Recently, Kheradmand et al. introduced the notion of RNP (reflexive-nilpotents-property) rings by restricting the reflexive property to nilpotent elements. In this article, we study reflexive-nilpotents-property skewed by a ring endomorphism alpha and introduce the notion of alpha -skew RNP rings. We investigate various properties and extensions of these rings and also determine the structure of minimal noncommutative alpha -skew RNP rings.

The corepresentation ring of a pointed Hopf algebra of rank one

Let [Formula: see text] be an arbitrary pointed Hopf algebra of rank one and [Formula: see text] the group of group-like elements of [Formula: see text]. In this paper, we give the decomposition of a tensor product of finite dimensional indecomposable right [Formula: see text]-comodules into a direct sum of indecomposables. This enables us to describe the corepresentation ring of [Formula: see text] in terms of generators and relations. Such a ring is not commutative if [Formula: see text] is not abelian. We describe all nilpotent elements of the corepresentation ring of [Formula: see text] if [Formula: see text] is a finite abelian group or a particular Hamiltonian group. In this case, all nilpotent elements of the corepresentation ring form a principal ideal which is either zero or generated by a nilpotent element of degree 2.

Rings in Which Nilpotent Elements Belong to Socle

Prompted by the notion of J-reduced rings in which the nilpotent elements belong to the Jacobson radical, we introduce right (left) \({\mathfrak {S}}\)-reduced rings as those satisfying the condition that the nilpotent elements reside in the right (left) socle. It turns out that a ring R is right (left) \({\mathfrak {S}}\)-reduced if and only if the quotient ring of R modulo its right (left) socle is reduced. Among other things, we shall determine exactly when the idealization of a module over a commutative ring as well as the formal triangular matrix rings has this property.

A discrete-time dynamical system and an evolution algebra of mosquito population.

Recently, continuous-time dynamical systems of mosquito populations have been studied. In this paper, we consider a discrete-time dynamical system, generated by an evolution quadratic operator of a mosquito population, and show that this system has two fixed points, which become saddle points under some conditions on the parameters of the system. We construct an evolution algebra, taking its matrix of structural constants equal to the Jacobian of the quadratic operator at a fixed point. Idempotent and absolute nilpotent elements, simplicity properties, and some limit points of the evolution operator corresponding to the evolution algebra are studied. We give some biological interpretations of our results.

Minimally ramified deformations when

Let $p$ and $\ell$ be distinct primes, and let $\overline{\unicode[STIX]{x1D70C}}$ be an orthogonal or symplectic representation of the absolute Galois group of an $\ell$-adic field over a finite field of characteristic $p$. We define and study a liftable deformation condition of lifts of $\overline{\unicode[STIX]{x1D70C}}$ ‘ramified no worse than $\overline{\unicode[STIX]{x1D70C}}$’, generalizing the minimally ramified deformation condition for $\operatorname{GL}_{n}$ studied in Clozel et al. [Automorphy for some$l$-adic lifts of automorphic mod$l$Galois representations, Publ. Math. Inst. Hautes Études Sci. 108 (2008), 1–181; MR 2470687 (2010j:11082)]. The key insight is to restrict to deformations where an associated unipotent element does not change type when deforming. This requires an understanding of nilpotent orbits and centralizers of nilpotent elements in the relative situation, not just over fields.