Let D be a division ring with the center F and suppose that D* is the multiplicative group of D. D is called centrally finite if D is a finite dimensional vector space over F and D is locally centrally finite if every finite subset of D generates over F a division subring which is a finite dimensional vector space over F. We say that D is a linear division ring if every finite subset of D generates over Fa centrally finite division subring. It is obvious that every locally centrally finite division ring is linear. In this report we show that the inverse is not true by giving an example of a linear division ring which is not locally centrally finite. Further, we give some properties of subgroups in linear division rings. In particular, we show that every finitely generated subnormal subgroup in a linear ring is central. An interesting corollary is obtained as the following: If D is a linear division ring and D* is finitely generated, then D is a finite field.

Let D be a division ring with the center F. We say that N is a subgroup of D with understanding that N is in fact a subgroup of the multiplicative group D* of D. In this note we disscus the conjecture which was posed by Herstein in 1978 [2, Conjecture 3]: If N is a subnormal subgroup of D which is radical over F, then N is contained in F. In his paper, Herstein himself showed that the conjecture is true if N is a finite subnormal subgroup of D. However, it is not proven for the general cases. In this note, we establish some properties of subnormal subgroups in division rings which could give some information in the direction of verifying this longstanding conjecture. In particular, it is shown that the conjecture is true for locally centrally finite division rings.