James Osterburg

Semiprime ideals in rings with finite group actions

LetR be a ring, G a finite group of automorphisms acting on R, and RG the-fixed subring of R. We prove that if R is semiprime with no additive ¦ G¦-torsion, then R is left Goldie if and only if RG is left Goldie. By coupling this with an examination of the prime ideal structures of RG and R, we are able to prove that if ¦G ¦ is invertible in R and RG is left Noetherian, then R satisfies the-ascending chain condition on semiprime ideals, every semiprime factor ring of R is left Goldie, and nil subrings of R are nilpotent. For the pair RG and R, we also consider various other properties of prime and maximal ideals such as lying over, going up, going down, and incomparability.

A Noether Skolem theorem for group-graded rings

Abstract The classical Noether Skolem theorem gives conditions to determine if an automorphism or a derivation of a finite-dimensional central simple algebra is inner. The definition of inner has been extended to more general Hopf algebra actions. In this paper, we study group-gradings of central simple algebras and we give sufficient conditions for the grading to be inner. This provides a partial answer to a question of J. Bergen and S. Montgomery.

X-inner automorphisms of enveloping rings

Abstract In this paper, we determine the X-inner automorphisms of the smash product R # U(L) of a prime ring R by the universal enveloping algebra U(L) of a characteristic 0 Lie algebra L. Specifically, we show that any such automorphism σ stabilizing R can be written as a product σ = σ1σ2, where σ1 is induced by conjugation by a unit of Q3(R), the symmetric Martindale ring of quotients of R, and σ2 is induced by conjugation by a unit of Q3(T). Here S = Ql(R) is the left Martindale ring of quotients of R and T is the centralizer of S in S # U(L) ⊃- R # U(L). One of the subtleties of the proof is that we must work in several unrelated overrings of R # U(L).

Computing the connes spectrum of a Hopf algebra

LetH be a finite-dimensional Hopf algebra over the fieldK and letA be anH-module algebra. In a previous paper, we defined the Connes spectrum CS(A, H) for the action ofH onA to be a certain subset of the set Irr(H) of irreducible representations ofH. In this paper, we compute a number of examples; specifically, we consider certain inner and outer actions and we take a closer look at the cocommutative situation. We discover that the information encoded in the Connes spectrum is rather subtle and elusive.

A strong Connes spectrum for finite group actions of simple rings

Abstract In this paper, we give necessary and sufficient conditions for a skew group ring to be simple. Our conditions are in terms of the irreducible representations of the group G. We also introduce a version of the strong Connes spectrum for finite group actions of simple rings.

A semiprime morita context related to finite automorphism groups of rings

A Morita context relating the fixed ring and the skew group ring introduced by M. Cohen is studied. If the skew group ring is semiprime andRG satisfied a PI, thenR satisfies a PI of degree ≦|G|d. We also discuss the Galois correspondence for the maximal quotient ring of a free algebra.

The Zeros and the Final Value of a Polynomial Form

ABSTRACT A polynomial form f, is a not necessarily linear map, from an infinite module over a ring 𝔷 to a finite abelian group of exponent n satisfying some additional conditions. Denote the zeros of f by Ωf. We show it satisfies a weak closure condition. Among all 𝔷-submodules of finite index, there is a submodule B such that |f (B)| (the order of the subset f (B)) is as small as possible. f (B) is called the final value of f and D. S. Passman asks if f (B) is necessarily a subgroup of S. This paper shows that if the degree of f ≤ 2 then the final value is a subgroup and if the form f has arbitrary degree from an finitely generated infinite abelian group, then the final value is 0. Added in Proof: D. S. Passmam has recently found a counterexample to the final value problem.

COCYCLE EQUIVALENT HOPF ALGEBRA ACTIONS

Let H be a Hopf algebra over a field K and assume a K-algebra A is an H-module algebra under two actions; · and ∘. We call these actions cocycle equivalent if there is an action of H on M 2(A), h ♦ X, such that for h ∈ H, X ∈ M 2(A) and a,b ∈ A. Two actions are cocycle equivalent if and only if there are cocycles that relate the two actions. Using these, it is shown that cocycle equivalence is a equivalence relation. Finally let H be a finite dimensional, semisimple, cocommutative Hopf algebra and assume K is a splitting field of H. It is shown that the Connes spectrum of H acting on M 2(A) is the intersection of the Connes spectra of H acting on A under · and ∘. Denote the smash product of A and H under the action · by (A#H,·). Let A be H-prime, then (A#H,·) is prime if and only if (A)#H, ♦) is prime if and only if (M 2(A)# H, ♦) is prime.

The Connes spectrum of group actions and group gradings for certain quotient rings

Let H be a finite-dimensional, semisimple Hopf algebra over an algebraically closed field K where H is either commutative or cocommutative. We let A be an //-module algebra which is semiprime right Goldie. We show that the Connes spectrum of H acting on A is the Connes spectrum of H acting on the classical quotient ring of A. In our last section, we define a symmetric quotient ring and show that the Connes spectrum of the ring and its quotient ring are the same. Finally, we apply our results to finite group actions and group gradings.

Fullness of connes spectra and the connes hopf kernel

Let H be a finite dimensional, semisimple Hopf algebra over a field K and let A be an H- module algebra. Assume K is a splitting field for H and that H is strongly semiprime. If A is H- semiprime, we show the Connes spectrum of H acting on A consists of all of the irreducible representations of H is equivalent to every nonzero annihilator ideal of the smash product meets A nontrivially. If H is also cocommutative, we let I′ be the intersection of the annihilators of the modules in the Connes spectrum. We find some of the information encoded in the Hopf kernel of the natural map from H to H/I′.