On Nilpotent Elements of Skew Polynomial Rings

We study the structure of the set of nilpotent elements in skew polynomial ring r[x; α], when r is an α-armendariz ring. we prove that if r is a nil α-armendariz ring and αt = ir, then the set of nilpotent elements of r is an α-compatible subrng of r. also, it is shown that if r is an α-armendariz ring and αt = ir, then r is nil α-armendariz. we give some examples of non α-armendariz rings which are nil α-armendariz. moreover, we show that if αt = ir for some positive integer t and r is a nil α-armendariz ring and nil(r[x][y; α]) = nil(r[x])[y], then r[x] is nil α-armendariz. some results of [3] follow as consequences of our results