Nilpotent Elements Research Articles

On extensions of the Jacobson–Morozov theorem to even characteristic

AbstractLet be a simple algebraic group over an algebraically closed field of characteristic 2. We consider analogues of the Jacobson–Morozov theorem in this setting. More precisely, we classify those nilpotent elements with a simple 3‐dimensional Lie overalgebra in and also those with overalgebras isomorphic to the algebras and . This leads us to calculate the dimension of the Lie automiser for all nilpotent orbits; in even characteristic this quantity is very sensitive to isogeny.

Modules with reduced endomorphism rings

In this paper, we study endo-reduced modules as modules whose endomorphism rings have no nonzero nilpotent elements. We characterize their properties for different classes of modules, including [Formula: see text]-non-singular modules, multiplication modules and finitely generated modules over commutative Dedekind domains. In the subcategory of finitely generated modules, it is shown that the class of rings [Formula: see text] for which every faithful multiplication [Formula: see text]-module is endo-reduced is precisely that of reduced rings; while the class of rings [Formula: see text] for which every multiplication [Formula: see text]-module is endo-reduced is precisely that of von Neumann regular rings. Characterizations of when an endo-reduced module will be a reduced module are given. We prove that a finitely generated module over a principal ideal domain (PID) is endo-reduced exactly if it is either a semisimple module with pair-wise non-isomorphic submodules or a torsion-free module which is isomorphic to the underlying ring.

A discussion of bisexual populations with Wolbachia infection as an evolution algebra

In this paper, Wolbachia infection in a bisexual and diploid population with a fixed cytoplasmic incompatibility rate w and maternal transmission rate d is studied as an evolution algebra. As the cytoplasmic incompatibility (CI) of the population causes deaths in the offspring, the evolution algebra of this model is not baric, and is a dibaric algebra if and only if the cytoplasmic incompatibility rate w is 1 and d = 1 . The idempotent elements are given in terms of d and w. Moreover, this algebra has no absolute nilpotent elements when CI expression w ≠ 1 .

On Nilpotent Elements and Armendariz Modules

For a left module MR over a non-commutative ring R, the notion for the class of nilpotent elements (nilR(M)) was first introduced and studied by Sevviiri and Groenewald in 2014 (Commun. Algebra, 42, 571–577). Moreover, Armendariz and semicommutative modules are generalizations of reduced modules and nilR(M)=0 in the case of reduced modules. Thus, the nilpotent class plays a vital role in these modules. Motivated by this, we present the concept of nil-Armendariz modules as a generalization of reduced modules and a refinement of Armendariz modules, focusing on the class of nilpotent elements. Further, we demonstrate that the quotient module M/N is nil-Armendariz if and only if N is within the nilpotent class of MR. Additionally, we establish that the matrix module Mn(M) is nil-Armendariz over Mn(R) and explore conditions under which nilpotent classes form submodules. Finally, we prove that nil-Armendariz modules remain closed under localization.

Direct and ordinal products realized by triangular norm operators with no zero divisors

In this note we continue the work of Chon, as well as Mezzomo, Bedregal, and Santiago, by studying algebraic operations on fuzzy posets and bounded fuzzy lattices. We first prove that fuzzy posets are closed under finite direct products whenever the triangular norm realizing the product construction has no zero divisors. This result is then extended to the case of bounded fuzzy lattices. Some immediate consequences are then obtained within the setting of direct products realized by triangular norms with no nilpotent elements as well as strictly monotone and cancellative triangular norms. We then introduce a triangular norm based construction of ordinal products and similarly show that fuzzy posets are closed under ordinal products whenever the triangular norm realizing the product construction has no zero divisors.

A note on splittable linear Lie algebras

A linear Lie algebra is splittable if it contains the semisimple and nilpotent parts of each element. It is early known that a solvable linear Lie algebra g is splittable if and only if g=a+n, where a is an abelian subalgebra of g composed of semisimple elements and n is the ideal of all nilpotent matrices of g. In this paper, using elementary linear algebra we give a direct proof of the theorem and related results. Besides, we determine the structure of linear Lie algebras composed of semisimple or nilpotent elements.

On finite sums of projections and Dixmier's averaging theorem for type II1 factors

Let M be a type II1 factor and let τ be the faithful normal tracial state on M. In this paper, we prove that given an X∈M, X=X⁎, then there is a decomposition of the identity into N∈N mutually orthogonal nonzero projections Ej∈M, I=∑j=1NEj, such that EjXEj=τ(X)Ej for all j=1,…,N. Equivalently, there is a unitary operator U∈M with UN=I and 1N∑j=0N−1U⁎jXUj=τ(X)I. As the first application, we prove that a positive operator A∈M can be written as a finite sum of projections in M if and only if τ(A)≥τ(RA), where RA is the range projection of A. This result answers affirmatively Question 6.7 of [9]. As the second application, we show that if X∈M, X=X⁎ and τ(X)=0, then there exists a nilpotent element Z∈M such that X is the real part of Z. This result answers affirmatively Question 1.1 of [4]. As the third application, we show that for X1,…,Xn∈M, there exist unitary operators U1,…,Uk∈M such that 1k∑i=1kUi⁎XjUi=τ(Xj)I,∀1≤j≤n. This result is a stronger version of Dixmier's averaging theorem for type II1 factors.

Right Central CNZ Property Skewed by Ring Endomorphisms

The concept of the reversible ring property concerning nilpotent elements was introduced by A.M. Abdul-Jabbar and C. A. Ahmed, who introduced the concept of commutativity of nilpotent elements at zero, termed as a CNZ ring, as an extension of reversible rings. In this paper, we extend the CNZ property through the influence of a central ring endomorphism alpha , introducing a new type of ring called a right alpha -skew central CNZ ring. This concept not only expands upon CNZ rings but also serves as a generalization of right alpha -skew central reversible rings. We explore various properties of these rings and delve into extensions of right alpha -skew central CNZ rings, along with examining several established results, which emerge as corollaries of our findings.

Some results on nil-injective rings

Let R be a ring. A right R-module is called nil-injective if for any element w is belong to the set of nilpotent elements, and any right R-homomorphism can be extended to R to M. If RR is nil-injective, then R is called a right nil-injective ring. A right R-module is called Wnil-injective if for each non-zero nilpotent element w of R, there exists a positive integer n such that wn not zero that right R-homomorphism f:wnR to M can be extended to R to M. If RR is right Wnil-injective, then is called a right Wnil-injective ring. In the present work, we discuss some characterizations and properties of right nil-injective and Wnil-injective rings.

A note on the Jacobson radical of Ore extensions

Let R be a ring with a monomorphism α and an α-derivation δ. In this article, we give a simple and different proof about the semiprimitivity of Ore extensions which states that the skew polynomial ring R [ x ; α , δ ] is semiprimitive reduced if and only if R is α-rigid. This unifies and extends a number of known results on the Jacobson radical in the special cases. Also, as an application of our results, by imposing constraints on α and δ, we completely identify the Jacobson radical of rings whose the set of all nilpotent elements has special conditions. Important examples are provided to illustrate the applications of the results.

Matsuo algebras in characteristic 2

We extend the theory of Matsuo algebras, which are certain non-associative algebras related to 3-transposition groups, to characteristic 2. The decompositions of our algebras are now induced by nilpotent elements associated to lines in the corresponding Fischer space, rather than idempotent elements associated to points. For many 3-transposition groups, this still gives rise to a Z/2Z-graded fusion law, and we provide a complete classification of when this occurs. In one particular small case, arising from the 3-transposition group Sym(4), the fusion law is even stronger, and the resulting Miyamoto group is an algebraic group Ga2⋊Gm.

Primary Submaximal Ideals in Commutative Weak Idempotent Rings

Let R be a commutative weak idempotent ring (cWIR, for short) with unity, N and be the set of all nilpotent and idempotent elements of R respectively. In this paper, we study the structure of primary submaximal ideals in R and prove that, if P is a primary submaximal ideal of R and for some , then is a maximal ideal of R and , where .

Idempotent and nilpotent elements in octonion rings over Z

In this paper, we show that the set O/Zp, where p is a prime number, does not form a skew field and discuss idempotent and nilpotent elements in the (finite) ring O/Zp. We provide examples and establish conditions for idempotency and nilpotency. Mathematics Subject Classification (2010): 15A33, 15A30, 20H25, 15A03. Received 27 July 2021; Accepted 14 December 2021

On Right CNZ Rings with Involution

The object of this paper is to present the notion of right CNZ rings with involutions, or, in short, right *-CNZ rings which are a generalization of right *-reversible rings and an extended of CNZ property . A ring R with involution * is called right *-CNZ if for any nilpotent elements x, y є R, xy = 0 implies yx * = 0. Every right *-CNZ ring with unity involution is CNZ but the converse need not be true in general, even for the commutative rings. In this note, we discussed some properties right *-CNZ ring. After that we explored right *-CNZ property on the extensions and localizations of the ring R.

Rings of nilpotent elements of monomial derivations on polynomial rings

We study the rings Nil D of nilpotent elements of R-derivations D on polynomial rings over a UFD R containing Q . The main result of this paper determines Nil D of the monomial derivations D on R [ x , y ] . Furthermore, as its application, we prove that, for each positive integer m, there exists a monomial R-derivation with a slice on R [ x , y , z ] such that the minimal number of generators of its kernel equals m.

Generalizations of quasielliptic curves

We generalize the notion of quasielliptic curves, which have infinitesimal symmetries and exist only in characteristic two and three, to a remarkable hierarchy of regular curves having infinitesimal symmetries, defined in all characteristics and having higher genera. This relies on the study of certain infinitesimal group schemes acting on the affine line and certain compactifications. The group schemes are defined in terms of invertible additive polynomials over rings with nilpotent elements, and the compactification is constructed with the theory of numerical semigroups. The existence of regular twisted forms relies on Brion's recent theory of equivariant normalization. Furthermore, extending results of Serre from the realm of group cohomology, we describe non-abelian cohomology for semidirect products, to compute in special cases the collection of all twisted forms.

The reduced ring of the 𝑅𝑂(𝐶₂)-graded 𝐶₂-equivariant stable stems

We describe in terms of generators and relations the ring structure of the R O ( C 2 ) RO(C_2) -graded C 2 C_2 -equivariant stable stems π ⋆ C 2 \pi _\star ^{C_2} modulo the ideal of all nilpotent elements. As a consequence, we also record the ring structure of the homotopy groups of the rational C 2 C_2 -equivariant sphere π ⋆ C 2 ( S Q ) \pi _\star ^{C_2}(\mathbb {S}_\mathbb {Q}) .

M-rnc rings

In this article, an element w of an associative ring R is called m-regular nil clean or m-rnc if expressed as w = am + b where am is m-regular element and b is a nilpotent element. R is named m-regular nil clean ring or m-rnc ring. If all the elements of a ring R are m-rnc, some characteristics and basic properties of m-rnc rings are presented in this work.

The diameter of the nil-clean graph of $ \mathbb{Z}_n $

<p>Let $ R $ be a ring with identity. An element $ r $ was called to be nil-clean if $ r $ was a sum of an idempotent and a nilpotent element in $ R $. The nil-clean graph of $ R $ was a simple graph, denoted by $ G_{NC}(R) $, whose vertex set was $ R $, where two distinct vertices $ x $ and $ y $ were adjacent if, and only if, $ x+y $ was a nil-clean element of $ R $. In the absence of the condition that vertex $ x $ is not the same as $ y $, the graph defined in the same way was called the closed nil-clean graph of $ R $, which may contain loops, and was denoted by $ \overline{G_{NC}}(R) $. In this short note, we completely determine the diameter of $ G_{NC}(\mathbb{Z}_n) $.</p>

CHARACTERIZATIONS OF t2-REVERSIBLE RINGS

This article aims to investigate the ring theoretic structures of (strongly) t2-reversible ring using the concept of non-zero tripotent elements. A ring R is said to be t2-reversible if ab = 0 implies bat2 = 0 for all a,b ∈ R and t is a non-zero tripotent element of R. It is proved that R is a t2-reversible ring if and only if t2 is left semicentral and t2Rt2 is a reversible ring. We also introduce and establish several characteristics of strongly t2-reversible rings. It is proved that every strongly t2-reversible ring is also a t2-reversible ring but the converse need not be true. Moreover we call, R is a right (left) t2-reduced ring if N(R)t2 = 0 (t2N(R) = 0), where N(R) stands for the set of all nilpotent elements of R and we have established some of its properties.