  Algebra extensions A < B where A is a left B-module such that the B-action
extends the multiplication in A are ubiquitous. We encounter 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 paper we are extending R.Wisbauer's method of constructing the central
closure 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 algebra of A, then the
self-injective hull \hat{A} of A as B-module becomes an k-algebra provided A
does not contain any nilpotent B-stable ideals. We show that under certain
assumptions \hat{A} can be identified with a subalgebra of the Martindale
quotient ring of A. This construction is then applied to Hopf module algebras.
