Let K be a field and let G be a group. If G acts on an abelian group V , then it acts naturally on any group algebra K[V ], and we are concerned with classifying the G-stable ideals of K[V ]. In this paper, we consider a rather concrete situation. We take G to be an infinite locally finite simple group acting in a finitary manner on V . When G is a finitary version of a classical linear group, ...

Let F = F (X) be a free group with basis X and Z[t] be a ring of polynomials with integer coefficients in t. In this paper we develop a theory of (Z[t], X)-graphs a powerful tool in studying finitely generated fully residually free (limit) groups. This theory is based on the Kharlampovich-Myasnikov characterization of finitely generated fully residually free groups as subgroups of the Lyndon’s ...

In a recent paper by the authors, the associative and the Lie algebras of Weyl type A[D] = A⊗F[D] were introduced, where A is a commutative associative algebra with an identity element over a field F of any characteristic, and F[D] is the polynomial algebra of a commutative derivation subalgebra D of A. In the present paper, a class of the above associative and Lie algebras A[D] with F being a ...

Let S = k[x, ,..., x,] be a polynomial ring over a field and let A4 = @,*-a, M, be a finitely generated graded module; in the most interesting case A4 is an ideal of S. For a given natural number p, there is a great interest in the question: Can M be generated by (homogeneous) elements of degree <p? No simple answer, say in terms of the local cohomology of M, is known; but somewhat surprisingly...

In a recent paper by Zhao and the author, the Lie algebras A[D] = A⊗ IF [D] of Weyl type were defined and studied, where A is a commutative associative algebra with an identity element over a field IF of any characteristic, and IF [D] is the polynomial algebra of a commutative derivation subalgebra D of A. In the present paper, the 2-cocycles of a class of the above Lie algebras A[D] (which are...

We prove analogs of results of Cox and Cox/Schenk on the structure of certain ideals in the bigraded polynomial ring k[s, u; t, v].

Using the principle that characteristic polynomials of matrices obtained from elements of a reductive group G over Q typically have splitting field with Galois group isomorphic to the Weyl group of G, we construct an explicit monic integral polynomial of degree 240 whose splitting field has Galois group the Weyl group of the exceptional group of type E8.

Let R = K[x;σ] be a skew polynomial ring over a division ring K. We introduce the notion of derivatives of skew polynomial at scalars. An analogous definition of derivatives of commutative polynomials from K[x] as a function of K[x] → K[x] is not possible in a non-commutative case. This is the reason why we have to define the derivative of a skew polynomial at a scalar. Our definition is based ...

The concepts of Boolean metric space and convex combination are used to characterize polynomial maps An −→ Am in a class of commutative Von Neumann regular rings including p-rings, that we have called CFG-rings. In those rings, the study of the category of algebraic varieties (i.e. sets of solutions to a finite number of polynomial equations with polynomial maps as morphisms) is equivalent to t...