Daniyar Kozybaev, Leonid Makar-Limanov, Ualbai Umirbaev
We prove the Freiheitssatz for right-symmetric algebras and the decidability of the word problem for right-symmetric algebras with a single defining relation. We also prove that two generated subalgebras of free right-symmetric algebras are free and automorphisms of two generated free right-symmetric algebras are tame.
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Over a field of characteristic zero, we prove that the Freiheitssatz holds for brace algebras, the word problem for the brace algebras with a single defining relation is decidable, two generated subalgebras of free brace algebras are free, and that automorphisms of two generated free brace algebras are tame.
We prove the Freiheitssatz for Poisson algebras in characteristic zero. We also give a proof of the tameness of automorphisms for two generated free Poisson algebras and prove that an analogue of the commutator test theorem is equivalent to the two dimensional classical Jacobian conjecture using the Freiheitssatz and Jung's Theorem.
$\textbf{Theorem 1.2.}$ For a ring $A$, the following conditions are equivalent. $\textbf{1)}$ $A$ is a right automorphism-invariant right non-singular ring. $\textbf{2)}$ $A$ is a right automorphism-invariant regular ring. $\textbf{3)}$ $A=S\times T$, where $S$ is a right injective regular ring and $T$ is a strongly regular ring which contains all invertible elements of its maximal right ring of quotients. $\textbf{Theorem 1.5.}$ For a ring $A$ with right Goldie radical $G(A_A)$, the following conditions are equivalent. $\textbf{1)}$ $A/G(A_A)$ is a semiprime right Goldie ring. $\textbf{2)}$ …
Every automorphism-invariant right non-singular $A$-module is injective if and only if the factor ring of the ring $A$ with respect to its right Goldie radical is a right strongly semiprime ring.
A module is called automorphism-invariant if it is invariant under any automorphism of its injective hull. In [Algebras for which every indecomposable right module is invariant in its injective envelope, Pacific J. Math., vol. 31, no. 3 (1969), 655-658] Dickson and Fuller had shown that if $R$ is a finite-dimensional algebra over a field $\mathbb F$ with more than two elements then an indecomposable automorphism-invariant right $R$-module must be quasi-injective. In this paper we show that this result fails to hold if $\mathbb F$ is a field with two elements. Dickson and Fuller had further sho…
In this paper, we study rings having the property that every right ideal is automorphism-invariant. Such rings are called right $a$-rings. It is shown that (1) a right $a$-ring is a direct sum of a square-full semisimple artinian ring and a right square-free ring, (2) a ring $R$ is semisimple artinian if and only if the matrix ring $\mathbb{M}_n(R)$ for some $n>1$ is a right $a$-ring, (3) every right $a$-ring is stably-finite, (4) a right $a$-ring is von Neumann regular if and only if it is semiprime, and (5) a prime right $a$-ring is simple artinian. We also describe the structure of an indec…
We consider rings whose one-sided ideals are close to automorphism-invariant modules. We study rings in which every (finitely generated) right ideal is automorphism invariant and rings in which every right ideal is a finite direct sum of automorphism invariant ideals. Connections between these classes of rings, $q$-ring and $Σ$-$q$-rings are also considered
Let R be a Lie nilpotent algebra of index k over a field K of characteristic zero. If G is an n-element subgroup of Aut(R) of the K-automorphisms, then we prove that R is right integral over Fix(G) of degree n^k. In the presence of a primitive n-th root of unity e in K, for a K-automorphism d in Aut(R) with d^n=id, we prove that the skew polynomial algebra R[w,d] is right integral of degree n^k over Fix(d)[w^n].
It is proved, among other results, that a prime right nonsingular ring (in particular, a simple ring) $R$ is right self-injective if $R_R$ is invariant under automorphisms of its injective hull. This answers two questions raised by Singh and Srivastava, and Clark and Huynh. An example is given to show that this conclusion no longer holds when prime ring is replaced by semiprime ring in the above assumption. Also shown is that automorphism-invariant modules are precisely pseudo-injective modules, answering a recent question of Lee and Zhou. Furthermore, rings whose cyclic modules are automorphi…
An example of a non-triangulable locally nilpotent derivation and an example of a wild exponential automorphism of the free Novikov algebra $N\left\langle x,y,z\right\rangle$ in three variables $x,y,z$ over a field of characteristic zero are constructed.
Let $\BB$ be the Lie algebra of Block type with basis $\{L_{\a,i}|\,\a,i\in\Z, i\geq0\}$ and relations $[L_{\a,i},L_{\b,j}]=\left((\a-1)(j+1)-(\b-1)(i+1)\right)L_{\a+\b,i+j}$. In the present paper, the derivation algebra and the automorphism group of $\BB$ are explicitly described. In particular, it is shown that the outer derivation space is 1-dimensional and the inner automorphism group of $\BB$ is trivial.
Let $A$ be a ring and $φ$ its automorphism. It is proved that skew Laurent series ring $A((x,φ))$ is a right serial ring if and only if $A$ is a right serial right Artinian ring.
If $A$ is a ring with automorphism $φ$ and the skew Laurent series ring $A((x,φ))$ is a right semidistributive semilocal ring then $A$ is a right semidistributive right Artinian ring. The Laurent series ring $A((x))$ is a right semidistributive semilocal ring if and only if $A$ is a right semidistributive right Artinian ring. The study is supported by Russian Scientific Foundation (project 16-11-10013).
A module is called automorphism-invariant if it is invariant under any automorphism of its injective hull. Dickson and Fuller have shown that if $R$ is a finite-dimensional algebra over a field $\mathbb F$ with more than two elements then an indecomposable automorphism-invariant right $R$-module must be quasi-injective. In this note, we extend and simplify the proof of this result by showing that any automorphism-invariant module over an algebra over a field with more than two elements is quasi-injective. Our proof is based on the study of the additive unit structure of endomorphism rings.
Let $A_{1} := k [t, \partial ]$ be the first algebra over a field $k$ of characteristic zero. One can associate to each right ideal $I$ of $A_1$ its Stafford subgroup, which is a subgroup of $\Aut_k(A_1)$, the automorphism group of the ring $A_1$. In this article we show that each Stafford subgroup is equal to its normalizer. For that, we study when the Stafford subgroup of a right ideal of $A_1$ contains a given Stafford subgroup.
We define the concept of \emph{companion automorphism} of a Hopf algebra $H$ as an automorphism $σ:H \rightarrow H$: $σ^2=S^2$ --where $S$ denotes the antipode--. A Hopf algebra is said to be \emph{almost involutive} (AI) if it admits a companion automorphism that can be viewed as a special additional symmetry. We present examples and study some of the basic properties and constructions of AI-Hopf algebras centering the attention in the finite dimensional case. In particular we show that within the family of Hopf algebras of dimension smaller or equal than 15, only in dimension eight and twelv…
Let R be a ring with identity, (M;\leq) a commutative positive strictly ordered monoid and w_m an automorphism for each m \in M . The skew generalized power series ring R[[M,w]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal'cev Neumann Laurent series rings. If S\subset R is a multiplicative set, then R is called right S-Noetherian, if for each ideal I of R, Is \subseteq J\subseteq I for some s\in S and some finitely generated right ideal J . Unifying and generalizing a number of known results, we s…
The inversion formula is given for automorphisms of the Weyl algebras with polynomial coefficients over a field of characteristic zero. The theorem of Gabber on the degree of polynomial automorphism is extended. It is proved that any automorphism of the Weyl algebra with polynomial coefficients is determined by its left (right) faces (extending the result of J. H. McKay and S. S.-S. Wang for polynomial automorphisms).
A Rota-Baxter algebra $A_R$ is an algebra $A$ equipped with a distinguished Rota-Baxter operator $R$ on it. Rota-Baxter algebras are closely related to dendriform algebras introduced by Loday. In this paper, we first consider the non-abelian extension theory of Rota-Baxter algebras and classify them by introducing the non-abelian cohomology. Next, given a non-abelian extension $0 \rightarrow B_S \rightarrow E_U \rightarrow A_R \rightarrow 0$ of Rota-Baxter algebras, we construct the Wells type exact sequences and find their role in extending a Rota-Baxter automorphism $β\in \mathrm{Aut}(B_S)$ …
A complete description of subgroups in the general linear group over a semilocal ring containing the group of diagonal matrices was obtained by Z.I.Borewicz and N.A.Vavilov. It is shown in the present paper that a similar description holds for the intermediate subgroups of the group of all automorphisms of the lattice of right submodules of a free finite rank R-module over a simple Artinian ring containing the group consisting of those automorphisms which leave invariant an appropriate sublattice.
Non-abelian extensions of Rota-Baxter Lie algebras and inducibility of automorphisms
本文研究了 Rota-Baxter 李代数的非阿贝尔扩张，定义了对其进行分类的非阿贝尔上同调，揭示了自同构可诱导性的障碍，并导出了 Wells 正合序列。
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