On nilpotent elements in a nearring of polynomials. (Q2915738)

For a ring \(R\) (with 1), \(R[x]=(R[x],+,\circ)\) will denote the (left) nearring of all polynomials over \(R\) with respect to the addition and composition of polynomials. \(R_0[x]\) is the subnearring of all zero-symmetric polynomials of \(R[x]\). The author investigates relationships between the set of nilpotent elements of the ring \(R\) and those of the nearring \(R[x]\). If \(\text{nil}(N)\) denotes the set of nilpotent elements of \(N\), \(N\) a ring or nearring, one of the main results is: Whenever \(\text{nil}(R)\) is a locally nilpotent ideal of the ring \(R\), then \(\text{nil}(R[x],+,\circ)=\text{nil}(R)_0[x]\) and it is a right ideal of the nearring \(R[x]\).