Symmetric Martindale Quotient Ring Research Articles

Biinvertible actions of Hopf algebras

We study actions of a Hopf algebraH on an algebraR such that the action is twisted by an invertible mapσ:H⊗H →R; the biinvertible condition means that these actions also have both an inverse and an antiinverse in Hom(H, EndR). WhenR is an ordinaryH-module algebra, the action is biinvertible if the antipose is bijective. As a new example we show that if theH-action is twisted and the coradical ofH is cocommutative, then the action is biinvertible. After studying the continuity of these actions with respect to the filter of ideals ofR with zero annihilator, we consider when the actions may be extended to the symmetric Martindale quotient ring ofR and itsH-analog. Our results can be applied to crossed productsR# σ .

X-inner automorphisms of enveloping rings

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 = σ 1 σ 2, where σ 1 is induced by conjugation by a unit of Q 3 ( R), the symmetric Martindale ring of quotients of R, and σ 2 is induced by conjugation by a unit of Q 3 ( T). Here S = Q l ( 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 symmetric ring of quotients

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