In 1993, Muzychuk \cite{muzychuk} showed that the rational Schur rings over a cyclic group $Z_n$ are in one-to-one correspondence with sublattices of the divisor lattice of $n$, or equivalently, with sublattices of the lattice of subgroups of $Z_n$. This can easily be extended to show that for any finite group $G$, sublattices of the lattice of characteristic subgroups of $G$ give rise to rational Schur rings over $G$ in a natural way. Our main result is that any finite group may be represented as the (algebraic) automorphism group of such a rational Schur ring over an abelian $p$-group, for any odd prime $p$. In contrast, we show that over a cyclic group the automorphism group of any Schur ring is abelian. We also prove a converse to the well-known result of Muzychuk \cite{muzychuk2} that two Schur rings over a cyclic group are isomorphic if and only if they coincide; namely, we show that over a group which is not cyclic, there always exist distinct isomorphic Schur rings.
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
We study automorphism groups of formal matrix algebras. We also consider automorphisms of ordinary matrix algebras (in particular, triangular matrix algebras).
Automorphism groups of direct products of multiplicative monoids of certain rings
University of Stellenbosch（斯泰伦博斯大学）；NITheCS (National Institute for Theoretical and Computational Sciences)（国家理论与计算科学研究所）；University of Minho（米尼奥大学）；CMAT (Centre of Mathematics)（数学中心）
Let R be a perfect ring of characteristic p. We show that the group of continuous R-linear automorphisms of the perfect power series ring over R is generated by the automorphisms of the ordinary power series ring together with Frobenius; this answers a question of Jared Weinstein.
Anderson和D 'Ambrosia将高维无平方代数完全刻画为系数在域K中的平方-自由半群S上的扭半群代数. D 'Ambrosia将无平方环的定义推广到了有单位元的Artin环，并证明了每一个无平方环都有一个相伴的除环D和一个无平方自由半群S.证明了无平方环可刻画为系数在除环D中的无平方自由半群S上的扭半群环。对于每一个无平方环，都存在一个短正合序列，它将无平方环的外自同构连接到与S和D有关的某些上同调群。
In this paper, we describe all automorphism-liftable torsion modules over non-primitive hereditary Noetherian prime rings. We also study automorphism-liftable non-torsion modules over not necessarily commutative Dedekind prime rings
Rings of Invariant Module Type and Automorphism-Invariant Modules
一个模被称为自同构不变的，如果它在其内射壳的任何自同构下是不变的。在[Algebras for which every indecomable right modules is invariant in its injective envelope，Pacific J. Math.，第31卷，第3期（1969），655-658] Dickson和Fuller证明了如果$R$是一个域$\mathbb F$上的有限维代数，该域有两个以上的元素，那么一个不可分解的自同构不变右$R$ -模必须是拟内射的。在本文中，我们表明，这一结果不成立，如果$\mathbb F$是一个域的两个元素。Dickson和Fuller进一步证明了，如果$R$是一个域$\mathbb F$上的有限维代数，且该域有两个以上的元素，那么$R$是右不变模类型当且仅当每个不可分解的右$R$ -模都是自同构不变的。本文将Dickson和Fuller的结果推广到任意的右Artin环.一个环$R$称为右自同构不变型（简称RAI型），如果每一个环生成的不可分解右$R$ -模都是自同构不变的.本文完全刻划了不可分解的RAI型右Artin环。
We define a nonassociative generalization of cyclic Azumaya algebras employing skew polynomial rings $D[t;σ]$, where $D$ is an Azumaya algebra of constant rank with center $C$ and $σ$ an automorphism of $D$, such that $σ|_{C}$ has finite order. The automorphisms of these algebras are canonically induced by ring automorphisms of the skew polynomial ring $D[t;σ]$ used in their construction. We achieve a description of their inner automorphisms. Results on the automorphisms of classical Azumaya algebras and central simple algebras of this type are obtained as special cases.
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].
We study actions by filtered automorphisms on classical generalized Weyl algebras (GWAs). In the case of a defining polynomial of degree two, we prove that the fixed ring under the action of a finite cyclic group of filtered automorphisms is again a classical GWA, extending a result of Jordan and Wells. Partial results are provided for the case of higher degree polynomials. In addition, we establish a version of Auslander's theorem for finite cyclic groups of filtered automorphisms acting on classical GWAs.
The left-ideal relation graph on a ring $R$, denoted by $\overrightarrow{Γ_{l-i}}(R)$, is a directed graph whose vertex set is all the elements of $R$ and there is a directed edge from $x$ to a distinct $y$ if and only if the left ideal generated by $x$, written as $[x]$, is properly contained in the left ideal generated by $y$. In this paper, the automorphisms of $\overrightarrow{Γ_{l-i}}(R)$ are characterized, where $R$ is the ring of all $n \times n$ matrices over a finite field $F_q$. The undirected left relation graph, denoted by $Γ_{l-i}(M_n(F_q))$, is the simple graph whose vertices are…
