Domination in the entire nilpotent element graph of a module over a commutative ring

Let R be a commutative ring with unity and M be a unitary R module. Let Nil(M) be the set of all nilpotent elements of M. The entire nilpotent element graph of M over R is an undirected graph E(G(M)) with vertex set as M and any two distinct vertices x and y are adjacent if and only if x + y ∈ Nil(M). In this paper we attempt to study the domination in the graph E(G(M)) and investigate the domination number as well as bondage number of E(G(M)) and its induced subgraphs N(G(M)) and Non(G(M)). Some domination parameters of E(G(M)) are also studied. It has been showed that E(G(M)) is excellent, domatically full and well covered under certain conditions.

Centers and Azumaya loci for finite W-algebras in positive characteristic

In this paper, we study the center Z of the finite W-algebra $${\mathcal{T}}({\mathfrak{g}},e)$$ associated with a semi-simple Lie algebra $$\mathfrak{g}$$ over an algebraically closed field $$\mathbb{k}$$ of characteristic p≫0, and an arbitrarily given nilpotent element $$e \in{\mathfrak{g}} $$ We obtain an analogue of Veldkamp’s theorem on the center. For the maximal spectrum Specm(Z), we show that its Azumaya locus coincides with its smooth locus of smooth points. The former locus reflects irreducible representations of maximal dimension for $${\mathcal{T}}({\mathfrak{g}},e)$$ .

INTEGRABILITY OF CLASSICAL AFFINE W-ALGEBRAS

We prove that all classical affine W-algebras 𝒲(𝔤; f), where g is a simple Lie algebra and f is its non-zero nilpotent element, admit an integrable hierarchy of bi-Hamiltonian PDEs, except possibly for one nilpotent conjugacy class in G2, one in F4, and five in E8.

A Remark on Quadrics in Projective Klingenberg Spaces over a Certain Local Algebra

This article is devoted to some polar properties of quadrics in the projective Klingenberg spaces over a local ring which is a linear algebra generated by one nilpotent element. In this case, polar subspaces are described; the notion “degree of neighborhood” is used for the geometric description of polar subspaces of quadrics. The polarity induced by a quadric is also studied.

A note on r-precious ring

An element of a ring [Formula: see text] is said to be [Formula: see text]-precious if it can be written as the sum of a von Neumann regular element, an idempotent element and a nilpotent element. If all the elements of a ring [Formula: see text] are [Formula: see text]-precious, then [Formula: see text] is called an [Formula: see text]-precious ring. We study some basic properties of [Formula: see text]-precious rings. We also characterize von Neumann regular elements in [Formula: see text] when [Formula: see text] is a Euclidean domain and by this argument, we produce elements that are [Formula: see text]-precious but either not [Formula: see text]-clean or not precious.

A note on the generalized nilpotent elements of a module

In this paper, we introduce the notion of the generalized nilpotent element of a module. In \cite{Groenewald}, the notion of nilpotent element of a module is introduced in the following sense: a non-zero element $m$ of an $R$-module $M$ is said to be nilpotent if there exists some $a\in R$ such that $a^k m=0$ but $am\neq 0$ for some $k\in \mathbb N$. In our present work we aim to generalize this notion. We have extended this notion to the strongly nilpotent element of a module.

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 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.