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We investigate 2-local Jordan automorphisms on operator algebras. In particular, we show that every 2-local Jordan automorphism of the algebra of all n× n real or complex matrices is either an automorphism or an anti-automorphism. The same is true for 2-local Jordan automorphisms of any subalgebra of ℬ which contains the ideal of all compact operators on X, where X is a real or complex separable Banach spaces and ℬ is the algebra of all bounded linear operators on X.
Let φ be a Jordan automorphism of an algebra . The situation when an element a ∈ satisfies is considered. The result which we obtain implies the Kleinecke-Shirokov theorem and Jacobson’s lemma.
The paper was motivated by Kovacs’ paper (1973), Isaacs’ paper (1980) and a recent paper, due to Brešar et al. (2018), concerning Skolem-Noether algebras. Let be a unital commutative ring, not necessarily a field. Given a unital -algebra , where is contained in the center of , , the goal of this paper is to study the question: when can a homomorphism be extended to an inner automorphism of ? As an application of main results presented in the paper, it is proved that if is a semilocal...
We connect the theorems of Rentschler [rR68] and Dixmier [jD68] on locally nilpotent derivations and automorphisms of the polynomial ring and of the Weyl algebra , both over a field of characteristic zero, by establishing the same type of results for the family of algebras
where is an arbitrary polynomial in . In the second part of the paper we consider a field of prime characteristic and study comodule algebra structures on . We also compute the Makar-Limanov invariant of absolute constants...
Let R be a ring with identity such that R⁺, the additive group of R, is torsion-free. If there is some R-module M such that and , we call R a Zassenhaus ring. Hans Zassenhaus showed in 1967 that whenever R⁺ is free of finite rank, then R is a Zassenhaus ring. We will show that if R⁺ is free of countable rank and each element of R is algebraic over ℚ, then R is a Zassenhaus ring. We will give an example showing that this restriction on R is needed. Moreover, we will show that a ring due to A....
The purpose of this paper is to prove the following result: Let be a -torsion free semiprime ring and let be an additive mapping, such that holds for all . In this case is left and right centralizer.
Let be a noncommutative prime ring of characteristic different from 2, with its two-sided Martindale quotient ring , the extended centroid of and . Suppose that is a nonzero -derivation of such that for all , where is an automorphism of , and are fixed positive integers. Then .
A ring R is called an E-ring if every endomorphism of R⁺, the additive group of R, is multiplication on the left by an element of R. This is a well known notion in the theory of abelian groups. We want to change the "E" as in endomorphisms to an "A" as in automorphisms: We define a ring to be an A-ring if every automorphism of R⁺ is multiplication on the left by some element of R. We show that many torsion-free finite rank (tffr) A-rings are actually E-rings. While we have an example of a mixed...
Let be the unique noncommutative and noncocommutative eight dimensional semi-simple Hopf algebra. We first construct a weak Hopf algebra based on , then we investigate the structure of the representation ring of . Finally, we prove that the automorphism group of is just isomorphic to , where is the dihedral group with order 12.
We introduce the notion of an automorphism liftable module and give a characterization to it. We prove that category equivalence preserves automorphism liftable. Furthermore, we characterize semisimple rings, perfect rings, hereditary rings and quasi-Frobenius rings by properties of automorphism liftable modules. Also, we study automorphism liftable modules with summand sum property (SSP) and summand intersection property (SIP).
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