Related papers: On one-sided Lie nilpotent ideals of associative r…

Let R be a commutative ring with identity. In this paper, we introduce the concept 1-absrbing primary ideal of R.

We show that there exist noncommutative Ore extensions in which every right ideal is two-sided. This answers a problem posed by Marks in Duo Rings and Ore extensions, J.Algebra 280(2), (2004). We also provide an easy construction of one…

For an arbitrary operator ideal I, every nilpotent element of I is a single commutator of operators from I^t, for an exponent t that depends on the degree of nilpotency.

Motivated by the concept of clean ideals, we introduce the notion of nil clean ideals of a ring. We define an ideal $I$ of a ring $R$ to be nil clean ideal if every element of $I$ can be written as a sum of an idempotent and a nilpotent…

This paper explores the properties of multiplicative Lie algebra structures on a nilpotent group of class $2$. We also present a method for determining a multiplicative Lie algebra structure on a group that serves as an extension of one Lie…

Let $\mathcal{N}_{\mathfrak{g}^*}$ be the variety of nilpotent elements in the dual of the Lie algebra of a reductive algebraic group over an algebraically closed field. In \cite{Lu2} Lusztig proposes a definition of a partition of…

Let L be a restricted Lie algebra over a field of characteristic p > 2 and denote by u(L) its restricted enveloping algebra. We determine the conditions under which the set of symmetric elements of u(L) with respect to the principal…

We study certain types of ideals in the standard Borel subalgebra of an untwisted affine Lie algebra. We classify these ideals in terms of the root combinatorics and give an explicit formula for the number of such ideals in type $A$. The…

We classify the nilpotent Lie algebras of real dimension eight and minimal center that admit a complex structure. Furthermore, for every such nilpotent Lie algebra $\mathfrak{g}$, we describe the space of complex structures on…

Commutative semirings with divisible additive semigroup are studied. We show that an additively divisible commutative semiring is idempotent, provided that it is finitely generated and torsion. In case that a one-generated additively…

An ideal $I$ of a ring $R$ is called left N-reflexive if for any $a\in$ nil$(R)$, $b\in R$, being $aRb \subseteq I$ implies $bRa \subseteq I$ where nil$(R)$ is the set of all nilpotent elements of $R$. The ring $R$ is called left…

An ideal $I$ of a commutative ring $R$ is said to be of linear type when its Rees algebra and symmetric algebra exhibit isomorphism. In this paper, we investigate the conjecture put forth by Jayanthan, Kumar, and Sarkar (2021) that if $G$…

We extend the classical construction of solvable Lie algebras from a nilradical to compatible Lie algebras. Since the sum of nilpotent ideals may fail to be nilpotent, we replace the usual nilradical by a \emph{special nilradical} that…

We continue the analysis of the weak commutativity construction for Lie algebras. This is the Lie algebra $\chi(\mathfrak{g})$ generated by two isomorphic copies $\mathfrak{g}$ and $\mathfrak{g}^{\psi}$ of a fixed Lie algebra, subject to…

Based upon properties of ordinal length, we introduce a new class of modules, the binary modules, and study their endomorphism ring. The nilpotent endomorphisms form a two-sided ideal, and after factoring this out, we get a commutative…

In this study, we present the generalization of the concept of $r$-ideals in commutative rings with nonzero identity. Let $R$ be a commutative ring with $0\neq1$ and $L(R)$ be the lattice of all ideals of $R$. Suppose that…

Semisimple Lie algebras have been completely classified by Cartan and Killing. The Levi theorem states that every finite dimensional Lie algebra is isomorphic to a semidirect sum of its largest solvable ideal and a semisimple Lie algebra.…

We consider associative algebras over a field. An algebra variety is said to be {\em Lie nilpotent} if it satisfies a polynomial identity of the kind $[x_1, x_2, ..., x_n] = 0$ where $[x_1,x_2] = x_1x_2 - x_2x_1$ and $[x_1, x_2, ..., x_n]$…

We prove that the nilpotent commuting variety of a reductive Lie algebra over an algebraically closed field of good characteristic is equidimensional. In characteristic zero, this confirms a conjecture of Vladimir Baranovsky. As a…

We study the minimal dimension of solvable and nilpotent Lie algebras over a field of characteristic zero with given derived length $k$. This is motivated by questions on nildecomposable Lie algebras $\Lg=\La+\Lb$, arising in the context of…