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Let R be a Noetherian domain and let (σ,δ) be a quasi-derivation of R such that σ is an automorphism. There is an induced quasi-derivation on the classical quotient ring Q of R. Suppose F=t
2−v is normal in the Ore extension R[t;σ,δ] where v∈R. We show F is prime in R[t;σ,δ] if and only if F is...
This paper surveys recent results on the problem of describing the equivalence classes of finite-dimensional central simple G-algebras over a field. Improvements to these results are provided in some situations.
This is an expository paper discussing a number of problems on eigenvalues of elements in representations of algebraic groups and finite Chevalley groups. We stress on the algebraic group case which relates also to similar problems for Lie groups and Lie algebras.
We say that an algebra R, graded by a group, is graded reversible if ab=0 implies ba=0 where a,b are homogeneous elements of R. In this note, we study graded reversibility when R=ℤG (viewed as a ℤ-algebra) and the grading group is cyclic of order two. A complete characterization of when this...
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.
In this survey paper we present recent classification results for gradings by arbitrary groups on finite-dimensional simple Lie algebras over an algebraically closed field of characteristic different from 2. We also describe the main tools that were used to obtain these results (in particular,...
We study the periodicity of the proper cocharacters and show that Regev’s conjecture holds in unitary algebras of P.I. exponent 2. Also we discuss the asymptotic behaviour of the codimensions and cocharacters of Clifford algebras and deal with other important examples of P.I. algebras.
A survey is given on recent results describing when a semigroup algebra K[S] of a submonoid S of a polycyclic-by-finite group is a prime Noetherian maximal order. As an application one constructs concrete classes of finitely presented algebras that have the listed properties. Also some open...
Let R be an associative ring with identity. An element a∈R is called clean if a=e+u with e an idempotent and u a unit of R and a is called strongly clean if, in addition, eu=ue. A ring R is called clean if every element of R is clean and R is strongly clean if every element of R is strongly...
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