$FC$-rings

Abstract

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.

Minimal non-BFC rings

“…(ii) This part it follows from [9, Lemma 2.2 and Theorem 2.1], [23, Proposition 1] and [6,Lemma 18(5)]. But we will prove it directly.…”

“…is a subgroup of finite index in the additive group R + of R [6]. In [9] such rings are called F IC.…”

Minimal non-BFC-rings

“…By analogy with the group theory (see e.g. [35,Chapther 14,14.5]), a ring R is called an F C-ring (or shortly F C) if, for any a ∈ R, the centralizer C R (a) := {c ∈ R | c • a = a • c} is a subgroup of finite index in the additive group R + of R [6]. In [9] such rings are called F IC.…”