Group rings satisfying generalized Engel conditions

Let R be a commutative ring with unity of characteristic r≥0 and G be a locally finite group. For each x and y in the group ring RG define [x,y]=xy-yx and inductively via [x ,_( n+1)  y]=[[x ,_( n)  y]  , y]. In this paper we show that necessary and sufficient conditions for RG to satisfies [x^m(x,y)   ,_( n(x,y))  y]=0 is: 1) if r is a power of a prime p, then G is a locally nilpotent group and G' is a p-group, 2) if r=0 or r is not a power of a prime, then G is abelian. In this paper, also, we define some generalized Engel conditions on groups, then we present a result about unit group of group algebras which satisfies this kind of generalized Engel conditions.