Commutative Associative Ring Research Articles

Автомофизмы нильтреугольных подколец алгебр Шевалле типа $G_2$ над областями целостности. I

Let $N\Phi(K)$ be the nil-triangular subalgebra of the Chevalley algebra over an associative commutative ring $K$ with the identity associated with a root system $\Phi$. This paper studies the well-known problem of describing automorphisms of Lie algebras and rings $N\Phi(K)$. Automorphisms of the Lie algebra $N\Phi(K)$ under restrictions $K=2K=3K$ on ring $K$ are described by Y. Cao, D. Jiang, J. Wang (2007). When passing from algebras to Lie rings, the group of automorphisms expands. Thus, the subgroup of central automorphisms is extended, i.e. acting modulo the center, ring automorphisms induced by automorphisms of the main ring are added. For the type $A_{n}$, a description of automorphisms of Lie rings $N\Phi(K)$ over $K$ was obtained by V.M. Levchuk (1983). Automorphisms of the Lie ring $N\Phi(K)$ are described by V.M. Levchuk (1990) for type $D_4$ over $K$, and for other types by A.V. Litavrin (2017), excluding types $G_2$ and $F_4$. In this paper we describe automorphisms of a nil-triangular Lie ring of type $G_2$ over an integrity domain $K$ under the following restrictions on $K$: either $K=2K=3K$, or $3K = 0$. To study automorphisms, the upper and lower central series described in this paper, are essentially used.

Автоморфизмы нильтреугольных подколец алгебр Шевалле типа G2 над полем характеристики 2

Let NΦ(K) be the nil-triangular subalgebra of the Chevalley algebra over an associative commutative ring K with the identity associated with a root system Φ (The basis of NΦ(K) consists of all elements er ∈ Φ+ of the Chevalley basis). This paper studies the well-known problem of describing automorphisms of Lie algebras and rings NΦ(K). Automorphisms of the Lie algebra NΦ(K) under restrictions K = 2K = 3K on ring K are described by Y. Cao, D. Jiang, J. Wang (Intern. J. Algebra and Computation, 2007). When passing from algebras to Lie rings, the group of automorphisms expands. Thus, the subgroup of central automorphisms is extended, i.e. acting modulo the center, ring automorphisms induced by automorphisms of the main ring are added. For the type An, a description of automorphisms of Lie rings NΦ(K) over K was obtained by V.M. Levchuk (Siberian Mathematical Journal, 1983). Automorphisms of the Lie ring NΦ(K) are described by V.M. Levchuk (Algebra and Logic, 1990) for type D4 over K, and for other types by A.V. Litavrin (Thesis for: Cand. Sc. (Physics and Mathematics) – 01.01.06. Siberian Federal University, 2017), excluding types G2 and F4. The author (2022) obtained a description of automorphisms of Lie rings NΦ(K) of type G2 when K is an integrity domain and K = 2K = 3K or 3K = 0. In this paper we describe automorphisms of a niltriangular Lie ring of type G2 over a field K under restriction 2K = 0. To study automorphisms, the upper and lower central series described in this work are essentially used. A new non-standard automorphism was found, called an S-automorphism.

Automorphisms and derivations of affine commutative and PI-algebras

We prove analogs of A. Selberg’s result for finitely generated subgroups of Aut ⁡ ( A ) \operatorname {Aut}(A) and of Engel’s theorem for subalgebras of Der ⁡ ( A ) \operatorname {Der}(A) for a finitely generated associative commutative algebra A A over an associative commutative ring. We prove also an analog of the theorem of W. Burnside and I. Schur about local finiteness of torsion subgroups of Aut ⁡ ( A ) \operatorname {Aut}(A) .

Two remarks on Lie rings of $2\times 2$ matrices over commutative associative rings

Let $C$ be an associative commutative ring with 1. If $a\in C$, then $aC$ denotes the principal ideal generated by $a$. Let $l, m, n$ be nonzero elements of $C$ such that $mn \in lC$. The set of matrices $\left( \begin{smallmatrix} a_{11} & a_{12} a_{21} & -a_{11} \end{smallmatrix} \right) $, where $a_{11}\in lC$, $a_{12}\in mC$, $a_{21}\in nC$, forms a Lie ring under Lie multiplication and matrix addition. The paper studies properties of these Lie rings.

Local Automorphisms of Nil-triangular Subalgebras of Classical Lie Type Chevalley Algebras

We study the problem of describing local automorphisms of nil-triangular subalgebra of the Chevalley algebra over an associative commutative ring with identity

Uniqueness of Addition in Lie Algebras of Chevalley Type over Rings with 1/2 and 1/3

In this paper, it is proved that Lie algebras of Chevalley type (An, Bn, Cn, Dn, E6, E7, E8, F4, and G2) over associative commutative rings with 1/2 (with 1/2 and 1/3 in the case of G2) have unique addition. As a corollary of this theorem, we note the uniqueness of addition in semisimple Lie algebras of Chevalley type over fields of characteristic ≠ 2 (≠ 2, 3 in the case of G2).

Derivations of Leavitt path algebras

In this paper, we describe the K-module HH1(LK(Γ)) of outer derivations of the Leavitt path algebra LK(Γ) of a row-finite graph Γ with coefficients in an associative commutative ring K with unit. We explicitly describe a set of generators of HH1(LK(Γ)) and relations among them. We also describe a Lie algebra structure of outer derivation algebra of the Toeplitz algebra. We prove that every derivation of a Leavitt path algebra can be extended to a derivation of the corresponding C⁎-algebra.

On a class of Lie rings of 2 × 2 matrices over associative commutative rings

Let k be an arbitrary associative and commutative ring. If are subgroups of the additive group of k such that , then matrices with form a subring of the Lie ring of all matrices over k whose trace is 0. The paper studies certain properties of these Lie rings. As an application of the notion introduced, the description of Lie rings lying between and , K an integral quadratic extension of , is given.

The Niltriangular Subalgebra of the Chevalley Algebra: the Enveloping Algebra, Ideals, and Automorphisms

The enveloping algebra of the niltriangular subalgebra NΦ(K) of the Chevalley algebra of type An−1 is the algebra of niltriangular n × n matrices over K. The enveloping algebras R of other types constructed so far are nonassociative. For classical types, an explicit description of automorphisms of the rings R over any commutative associative ring with an identity is given; in the case where K is a field, all ideals in R are also listed. The enumeration of ideals in R for K = GF(q) leads to a solution of a combinatorial problem concerning ideals of the algebras NΦ(K).

Isomorphisms of the linear groups GL2(R) over associative rings R

Isomorphisms of the linear groups GL2(R) over associative rings R with 1/2 and 1/3 are considered. In particular, we give a full description of automorphisms φ:GL2(R)→GL2(R), where R is any commutative associative ring with 1 and 1/2, 3 is non-zero-divisor, and R is generated by invertible elements.

On matrix Lie rings over an associative commutative ring with 1 that contain the special orthogonal Lie ring

Let K be an associative and commutative ring with 1, and k a subring of K such that and the elements 2, 3 are invertible in k. Let n be an integer and an invertible diagonal n by n matrix over k. In the paper, Lie subrings of the general linear Lie ring containing the special orthogonal Lie ring are described provided that .

ON ω-LOCAL MODULES AND Rad-SUPPLEMENTED MODULES

All modules considered in this note are over associative commutative rings with an identity element. We show that a <TEX>${\omega}$</TEX>-local module M is Rad-supplemented if and only if M/P(M) is a local module, where P(M) is the sum of all radical submodules of M. We prove that <TEX>${\omega}$</TEX>-local nonsmall submodules of a cyclic Rad-supplemented module are again Rad-supplemented. It is shown that commutative Noetherian rings over which every w-local Rad-supplemented module is supplemented are Artinian. We also prove that if a finitely generated Rad-supplemented module is cyclic or multiplication, then it is amply Rad-supplemented. We conclude the paper with a characterization of finitely generated amply Rad-supplemented left modules over any ring (not necessarily commutative).

Almost Commutative Varieties of Associative Rings and Algebras Over a Finite Field

Associative algebras over an associative commutative ring with unity are considered. A variety of algebras is said to be permutative if it satisfies an identity of the form x 1 x 2⋯x n = x 1σ x 2σ ⋯x nσ , where σ is a nontrivial permutation on a set {1, 2, … , n}. Minimal elements in the lattice of all nonpermutative varieties are called almost permutative varieties. By Zorn’s lemma, every nonpermutative variety contains an almost permutative variety as a subvariety. We describe almost permutative varieties of algebras over a finite field and almost commutative varieties of rings. In [5], such varieties were characterized for the case of algebras over an infinite field.

Local automorphisms of nilpotent algebras of matrices of small orders

Let K be an associative commutative ring with identity, and let R be the algebra of lower niltriangular n × n matrices over K. For n = 3, we prove that local automorphisms and Lie automorphisms of the algebra R generate all its local Lie automorphisms. For the case when K is a field and n = 4, we describe local automorphisms, local derivations, and local Lie automorphisms of the algebra R.

Rings with the greatest common divisor

We have found some class between the class of associative commutative factorial rings and the class of domains of integrity. It is the class of GCD-rings or almost factorial rings, i.e., domains of integrity where any two elements have a greatest common divisor.

Hereditarily pure associative algebras over a Dedekind ring whose maximal ideals have finite indices

The concepts of purity and of weak purity play an important part in Abelian group theory. It turns out that these concepts may be approached by using the theory of varieties of groups. The idea behind the approach allowed their analogs to be defined for arbitrary algebras and made it possible to formulate major problems of research in this direction (see [1]). Within these frames, the analogs mentioned were explored for modules [2-4], universal algebras [5-7], semigroups [8-12], Lie algebras over an arbitrary associative commutative ring with unity [13], and associative rings [14]. In so doing, more attention was centered on the concept of atomic purity, an analog of weak purity. In fact, atomic purity seems to be more natural to be dealt with in studying a good many algebras, at least at the initial stage. In the present paper, we will look at this concept as applied to associative algebras over a Dedekind ring. (In our argument, atomic purity will be referred to merely as purity.) Namely, we will work to tackle the problem of describing hereditarily pure algebras [1, Problem 17] for a variety of all algebras over a Dedekind ring R whose maximal ideals have finite indices. (In what follows, unless otherwise stated, we retain the designation R for a ring with the property specified.) Note that our reasoning covers the case of algebras over a finite field and the case of associative rings, and incidentally, refines a result announced in [14]. We give necessary definitions and designations. The letter p always stands for a prime, while k, m, and n are used to denote nonnegative integers. The cardinal number of a set M is denoted by |M |. Throughout the paper, a module is a left unitary R-module; an algebra is an associative

Some subsets of automata over a finite ring

Sets of Mealy and Moore automata over an arbitrary finite commutative-associative ring are investigated in which transition and output functions are linear combinations of functions of automaton states and functions of inputs. Subsets of strongly connected, reduced, and reversible automata and automata with permutation transition functions are characterized.

On the complexity of analysis of automata over a finite ring

A general scheme is investigated that is destined for obtaining estimates based on the cardinality of subsets of a fixed set of automata over some finite commutative-associative ring with unit element. A scheme is proposed to solve parametric systems of polynomial equations based on classes of associated elements of the ring. Some general characteristics of automata over the ring are established.

Reconstruction of the Initial State Vector of Nonlinear Automata over a Finite Ring

For Mealy and Moore automata over a finite commutative-associative ring with unity, which transition function is defined by nonlinear equations of the second degree, and the output function is affine and linear map of the states set, respectively, the problem of reconstruction of the initial state vector is solved. The case, when this problem is trivial, is considered. It is found that in other cases the problem is difficult. It is found that the property "to be reversible automaton" generally does not affect the complexity of solving the problem of reconstruction of the initial state vector for the investigated automata.

Invertible and Nilpotent Elements in the Group Algebra of a Unique Product Group

We describe the nilpotent and invertible elements in group algebras k[G] for k a commutative associative unital ring and G a unique product group, for example an ordered group.