Throughout the present paper, R will be a simple ring, where we shall understand by a simple ring a total matrix ring over a division rings. If S' is any subring containing the identity element 1 of R, we denote by VR(S') the centralizer of S' in R, VR(S') = VR(VR(S')) and by 0 (S', R) we denote the group of all automorphisms of R which are the identity on S'. We shall be concerned with a fixed S' which is denoted by S and shall consider primarily subrings S' of R which contain S. We abbreviate VR(S) = V, VR(S) = H, (S, R) = 0. A subring S' of R is said to be regular when S' and VR(S') are both simple. Further, if S' is regular and coincides with fixed ring J((O(S',R),R) of (S',R) in R, then we say that R is Galois over S'. In particular, if R is Galois over S and V coincides with the center of R then we say that R is outer Galois over S. If, for each finite subset F of R, the ring S[ F] generated by S and F is a finitely generated left S-module, then we say that R is left locally finite over S. If S* is a simple subring containing the identity element of R then, as is well known, any subring T of R containing S* contains a linearly independent left (right) basis over S*. By [ T: S*], ([ T: S*]r) denote the left (right) dimension. In case [ T: S*], = [ T: S*]r, then they are denoted by [ T: S*]. If M is any subset of R, we denote by Ml (Mr) the set of left (right) multiplications determined by elements of M. For any regular element a of R, we shall denote by (a) an inner automorphism ala-1 of R and by (M) denote the set of inner automorphisms determiined by regular elements of M. In our papers cited in the references, (M) has been denoted as M. We shall understand by a Hom(R, R)-module M a right Hom(R, R)module M. For any subset 0 of Hom(R, R), and for any subset M of R, we denote by a I M the restriction of a to M and by # (a I M) denote the cardinal number of a I M. We shall consider the following conditions: (Al): (i) S is regular and (5Rr is dense in Homs1(R, R) in the finite topology, and (ii) R is left locally finite over S. (Bl): (i) R is Galois over S and R is S, Vi-Rr-irreducible, and