Abstract. We investigate properties of F C -rings (i.e. rings R in which the centralizer C R .a/ of any element a 2 R is of finite index in R) and, in particular, characterize left Artinian rings with a finite set of all derivations Der R (respectively inner derivations IDer R). We show that if R is a Jacobson radical ring in which its adjoint group R ı has a finite number of conjugacy classes, thenis a ring direct sum of Jacobson radical rings R p i and D, where the additive group D C is a torsion-free divisible group, the adjoint group D ı is a group with a finite number of conjugacy classes, R C p i is a finite p i -group .i D 1; : : : ; t/ and p 1 ; : : : ; p t are pairwise distinct primes.