Let R be a commutative radical Fréchet algebra having a nonnilpotent element a with a ∈ Ra. Then R contains a continuum of incomparable prime ideals. In [3], J. Esterle proved the following result; it is a main ingredient in his proof that epimorphisms from C0(Ω) onto Banach algebras are continuous. Theorem (Esterle). Let R be a commutative radical Banach algebra. Suppose that there exists a no...

Let k be a commutative ring and let R be a commutative k−algebra. Given a positive integer n and a R−algebra A one can consider three functors of points from the category CR of commutative R−algebras to the small category of sets. All these functors are representable, namely • RepA represents the functor induced by B → homR(A,Mn(B)), where Mn(B) are the n× n matrices over B, for all B ∈ CR. • t...

Overview. My work is in the area of commutative algebra, and is motivated by the connections between algebra and geometry. Commutative algebra is the study of commutative rings (e.g. polynomial rings over fields and their quotients) and modules over these rings, while algebraic geometry is the study of finitely many polynomial equations in finitely many unknowns. The set of all solutions to suc...

The concept of a Hopf algebra originated in topology. Classically, Hopf algebras are defined on the basis of unital modules over commutative, unital rings. The intention of the present work is to study Hopf algebra formalism (§1.2) from a universal-algebraic point of view, within the context of entropic varieties. In an entropic variety, the operations of each algebra are homomorphisms, and ten...

We describe the non–commutative extension of the computer algebra system Singular, called Plural. In the system, we provide rich functionality for symbolic computation within a wide class of non– commutative algebras. We discuss the computational objects of Plural, the implementation of main algorithms, various aspects of software engineering and numerous applications. Singular:Plural or, short...

An ideal I in the free associative algebra k〈X1, . . . ,Xn〉 over a field k is shown to have a finite Gröbner basis if the algebra defined by I is commutative; in characteristic 0 and generic coordinates the Gröbner basis may even be constructed by lifting a commutative Gröbner basis and adding commutators.

Let B be a strictly real commutative real Banach algebra with the carrier space Φ B. If A is a commutative real Banach algebra, then we give a representation of a ring homomorphism ρ : A → B, which needs not be linear nor continuous. If A is a commutative complex Banach algebra, then ρ(A) is contained in the radical of B. 1. Introduction and results. Ring homomorphisms are mappings between two ...

We give a simple characterization for a nonassociative algebra A, having characteristic 6= 2, to be commutative. Namely, A is commutative if and only if it is exible with a commuting set of generators. A counterexample shows that characteristic 6= 2 is necessary. Both the characterization and the counterexample were discovered using the computer algebra system in [2].

Let H be a Hopf algebra with a bijective antipode over a commutative ring k with unit. The Brauer group of H is defined as the Brauer group of Yetter–Drinfel’d H-module algebras, which generalizes the Brauer–Long group of a commutative and cocommutative Hopf algebra and those known Brauer groups of structured algebras.

There are discussed implementational aspects of the special-purpose computer algebra system FELIX designed for computations in constructive algebra. In particular, data types developed for the representation of and computation with commutative and non-commutative polynomials are described. Furthermore, comparisons of time and memory requirements of di erent polynomial representations are reported.