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In this article, we introduce the concepts of excision and idealization for a multiplicative Lie algebra (also for a Lie algebra), which provides two new multiplicative Lie algebras (or Lie algebras) from a given multiplicative Lie algebra…
In this paper, we introduce the concept of ideal on CL-algebra. It is proved that this concept generalizes the notion of ideal on Residuated Lattices. Prime ideal on CL-algebra are defined and few interesting properties are obtained. It has…
We prove that the lattice of ideals of an arbitrary $L$-algebra is distributive. As a consequence, a spectral theory applies with no restriction. We also study the spectrum (i.e. the set of prime ideals) of $L$-algebras and characterize…
We link the recent theory of $L$-algebras to previous notions of Universal Algebra and Categorical Algebra concerning subtractive varieties, commutators, multiplicative lattices, and their spectra. We show that the category of $L$-algebras…
In this work, we extend the definition of the graded prime ideals from those in commutative graded rings to the ideals over graded Lie algebras. We prove some facts about graded prime Lie ideals in arbitrary Lie algebras that are similar to…
In this paper using the connections between some subvarieties of residuated lattices, we investigated some properties of the lattice of ideals in commutative and unitary rings. We give new characterizations for commutative rings $A$ in…
The notion of a non-associative universal enveloping algebra for a Lie triple system arises when Lie triple systems are considered as Bol algebras (more generally, Sabinin algebras). In this paper a new construction for these universal…
In this article, we introduce a lattice congruence with respect to a nonempty ideal $I$ of a distributive lattice $L$ and a derivation $d$ on $L$ denoted by $\theta_I^d$. We investigate some necessary and sufficient conditions for the…
In this paper we shall consider the Lie algebra of column-finite infinite matrices indexed by positive integers $\mathbb{N}$, describe the lattice of its ideals for arbitrary field $K$ and study its derivations over any commutative, unital…
There is an extensive recent literature on the graded, non-graded, prime, primitive, maximal ideals of Leavitt path algebras. In this introductory level survey, we will be giving an overview of different types of ideals and the…
The purpose of this paper is twofold. Firstly, to emphasise that the class of Lie algebras with chain lattices of ideals are elementary blocks in the embedding or decomposition of Lie algebras with finite lattice of ideals. Secondly, to…
Infinite hyperplane arrangements whose vertices form a lattice are studied from the point of view of commutative algebra. The quotient of such an arrangement modulo the lattice action represents the minimal free resolution of the associated…
A classification up to automorphism of the inner ideals of the real finite-dimensional simple Lie algebras is given, jointly with precise descriptions in the case of the exceptional Lie algebras.
A constructive procedure is given to determine all ideals of a solvable Lie algebra. This is used in determining algorithmically all conjugacy classes of subalgebras of a given solvable Lie algebra.
A subalgebra B of a Lie algebra L is called a c-ideal of L if there is an ideal C of L such that L = B + C and B \cap C \leq B_L, where B_L is the largest ideal of $L$ contained in B. This is analogous to the concept of c-normal subgroup,…
In the paper, we introduce some stabilizers and investigate related properties of them in MTL-algebras.Then, we also characterize some special classes of MTL-algebras, for example, IMTL-algebras, integral MTL-algebras, G\"{o}del algebras…
Consider a grade 2 perfect ideal $I$ in $R=k[x_1,\cdots,x_d]$ which is generated by forms of the same degree. Assume that the presentation matrix $\varphi$ is almost linear, that is, all but the last column of $\varphi$ consist of entries…
A subspace H of a Leibniz algebra L is called a quasi-ideal if [H;K] + [K;H] \subseteq H + K for every subspace K of L. They include ideals and subalgebras of codimension one in L. Quasi-ideals of Lie algebras were classified in two…
In this paper we focus on the structure of the variety of Lie algebras with a finite number of ideals and their graph representations using Hasse diagrams. The large number of necessary conditions on the algebraic structure of this type of…
This paper presents a unified framework for determining the congruences on a number of monoids and categories of transformations, diagrams, matrices and braids, and on all their ideals. The key theoretical advances present an iterative…