Computing the symmetric ring of quotients

AbstractWe discuss and compute the symmetric Martindale ring of quotients for various classes of prime rings. In particular, we consider free algebras and group algebras

Rewritable groups

AbstractA group G is said to have the n-rewritable property Qn if for all elements g1,g2,…,gn∈G, there exist two distinct permutations σ,τ∈Symn such that gσ(1)gσ(2)⋯gσ(n)=gτ(1)gτ(2)⋯gτ(n). We show here that if G satisfies Qn, then G has a characteristic subgroup N such that |G:N| and |N′| are both finite and have sizes bounded by functions of n. This extends the result of Blyth (1988) in [3] which asserts that if G satisfies Qn and if Δ is the finite conjugate center of the group, then |G:Δ| and |Δ′| are both finite with |G:Δ| bounded by a function of n. As a consequence, any group with Qn satisfies the permutational property Pm with m bounded by a function of n

Group algebras and enveloping algebras with nonmatrix and semigroup identities

Let K be a field of positive characteristic p, let R be either a group algebra K[G] or a restricted enveloping algebra u(L), and let I be the augmentation ideal of R. We first characterize those R for which I satisfies a polynomial identity not satisfied by the algebra of all 2-by-2 matrices over K. Then we examine those R for which I satisfies a semigroup identity (that is, a polynomial identity which can be written as the difference of two monomials).Comment: 11 pages. Written in LaTeX2

A description of incidence rings of group automata

Group automata occur in the Krohn-Rhodes Decomposition Theorem and have been extensively investigated in the literature. The incidence rings of group automata were introduced by the first author in analogy with group rings and incidence rings of graphs. The main theorem of the present paper gives a complete description of the structure of incidence rings of group automata in terms of matrix rings over group rings and their natural modules. As a consequence, when the ground ring is a field, we can use known group algebra results to determine when the incidence algebra is prime, semiprime, Artinian or semisimple. We also offer sufficient conditions for the algebra to be semiprimitive

Computing the symmetric ring of quotients

AbstractWe discuss and compute the symmetric Martindale ring of quotients for various classes of prime rings. In particular, we consider free algebras and group algebras

Prime ideals in nilpotent Iwasawa algebras

Let G be a nilpotent complete p-valued group of finite rank and let k be a field of characteristic p. We prove that every faithful prime ideal of the Iwasawa algebra kG is controlled by the centre of G, and use this to show that the prime spectrum of kG is a disjoint union of commutative strata. We also show that every prime ideal of kG is completely prime. The key ingredient in the proof is the construction of a non-commutative valuation on certain filtered simple Artinian rings

Del Pezzo surfaces of degree 1 and jacobians

We construct absolutely simple jacobians of non-hyperelliptic genus 4 curves, using Del Pezzo surfaces of degree 1. This paper is a natural continuation of author's paper math.AG/0405156.Comment: 24 page

Fixed rings and integrality

AbstractIn this note, we offer some variants of a theorem of Paré and Schelter on integrality in matrix rings. We use these to deduce that a ring R is integral over the fixed subring RG under the action of a finite abelian group G provided ¦G¦ · R = R. We also obtain an integrality version of the Bergman-Isaacs theorem