INTERNATIONAL CONFERENCE ON ALGEBRA IN MEMORY OF KOSTIA BEIDAR
National Cheng Kung University, Tainan March 6–12, 2005
A CENTRAL CLOSURE CONSTRUCTION FOR
CERTAIN ALGEBRA EXTENSIONS
CHRISTIAN LOMP
PORTO, PORTUGAL
Algebra extensions A⊆B where A is a left B-module such that the B-action extends the multiplication in A are ubiquitous. We en- counter examples of such extensions in the study of group actions, group gradings or more general Hopf actions as well as in the study of the bimodule structure of an algebra. In this talk we describe how to extend Robert Wisbauer’s method of constructing the central clo- sure of a semiprime algebra using its multiplication algebra to those kinds of algebra extensions. More precisely if A is a k-algebra and B some subalgebra of End(A)that contains the multiplication alge- bra of A, then the self-injective hullA of A as B-module becomes ab k-algebra provided A does not contain any nilpotent B-stable ideals.
We show that under certain assumptionsA can be identified with ab subalgebra of the Martindale quotient ring of A. This construction is then applied to Hopf module algebras.
