  Any finite set of linear operators on an algebra $A$ yields an operator
algebra $B$ and a module structure on A, whose endomorphism ring is isomorphic
to a subring $A^B$ of certain invariant elements of $A$. We show that if $A$ is
a critically compressible left $B$-module, then the dimension of its
self-injective hull $A$ over the ring of fractions of $A^B$ is bounded by the
uniform dimension of $A$ and the number of linear operators generating $B$.
This extends a known result on irreducible Hopf actions and applies in
particular to weak Hopf action. Furthermore we prove necessary and sufficient
conditions for an algebra A to be critically compressible in the case of group
actions, group gradings and Lie actions.
