On Nilpotent Elements of Skew Polynomial Rings

Abstract

We study the structure of the set of nilpotent elementsin skew polynomial ring

R[x; ], when R is an
-Armendariz ring. We prove that if R is a nil -Armendariz ring and
t = IR, then the set of nilpotent elements of R is an
-compatible subrng of R. Also, it is shown that if R is an
-Armendariz ring and t = IR, then R is nil-Armendariz. We give some examples of non -Armendariz rings which are nil -Armendariz. Moreover, we show that if t = IR for some positive integer t and R is a nil -Armendariz ring and nil(R[x][y; ]) = nil (R[x])[y], then R[x] is nil -Armendariz. Some results of [3] followas consequences of our results.