Finite topologies and their applications in linear algebra

In this paper, using finite topologies defined on the algebra of linear operators, we investigate centralizers and double centralizers of locally algebraic linear operators. In particular, for an arbitrary locally algebraic operator $A$, we establish the conditions under which the equality $CC(A)=C(A)$ is fulfilled, and in the case of an algebraically closed field, we describe minimal locally algebraic linear operators. Besides, we have studied automorphisms of dense in finite topology subrings of the rings of endomorphisms of free modules over projectively free rings.