Let \(R\) be an associative ring with identity and let \({\mathcal P}(R)\), \(N_r(R)\) and \({\mathcal N}(R)\) denote the prime radical, the upper nilradical and the set of all nilpotent elements of \(R\), respectively. \(R\) is \(2\)-primal if \({\mathcal P}(R)={\mathcal N}(R)\) and \(R\) is reduced if \({\mathcal N}(R)=0\). A ring \(R\) is strongly prime if \(R\) is prime with no nonzero nil ideals. An ideal \(P\) of \(R\) is strongly prime if the factor-ring \(R/P\) is a strongly prime ring. An ideal \(P\) of \(R\) is minimal strongly prime if \(P\) is minimal among all strongly prime ideals of \(R\). The set of all (minimal) strongly prime ideals of \(R\) is denoted by \(\text{(m)Spect}_S(R)\). Let \(\sigma\) be an endomorphism of a ring \(R\). An ideal \(I\) of \(R\) is called a \(\sigma\)-ideal if \(\sigma(I)\subseteq I\), and \(I\) is called \(\sigma\)-invariant if \(\sigma^{-1}(I)=I\). \(\sigma\) is called rigid if \(a\sigma(a)=0\) implies \(a=0\) for \(a\in R\). \(R\) is called a \(\sigma\)-rigid ring if the endomorphism \(\sigma\) is rigid. In this paper the authors extend the \(\sigma\)-rigid property of a ring \(R\) to \(N_r(R)\) by introducing the condition (*): \(N_r(R)\) is a \(\sigma\)-ideal of \(R\) and \(a\sigma(a)\in N_r(R)\) implies \(a\in N_r(R)\) for \(a\in R\) and then \(N_r(R)\) is said to be \(\sigma\)-completely semiprime. The authors investigate properties of a ring \(R\) with an endomorphism \(\sigma\) satisfying property (*) as well as characterize the upper nilradical \(N_r(R[\![x;\sigma]\!])\) of the skew power ring \(R[\![x;\sigma]\!]\) of a ring \(R\) (that is, the ring obtained by giving the power series ring over \(R\) the new multiplication: \(xr=\sigma(r)x\)), using \(N_r(R)\). In particular, they show that: (1) If \(P\) is \(\sigma\)-invariant for each \(P\in\text{(m)Spect}_S(R)\), then \(N_r(R)\) is \(\sigma\)-invariant. (2) If \(P\) is a \(\sigma\)-ideal for each \(P\in\text{(m)Spect}_S(R)\), then \(N_r(R)\) is a \(\sigma\)-ideal. (3) If \(N_r(R)={\mathcal N}(R)\), then (i) \(N_r(R)\) is a \(\sigma\)-ideal of \(R\) and (ii) \(N_r(R)\) is \(\sigma\)-invariant, where \(\sigma\) is a monomorphism. The authors show that the converses of (1), (2) and (3)(i) do not hold and that the condition `\(\sigma\) is a monomorphism' in (3)(ii) is not superfluous. They conclude that if \(R\) is \(2\)-primal, then \({\mathcal P}(R)\) is a \(\sigma\)-ideal of \(R\) and \({\mathcal P}(R)\) is \(\sigma\)-invariant when \(\sigma\) is a monomorphism. The authors note that if \(R\) is a \(\sigma\)-rigid ring, then \(N_r(R)\) is a \(\sigma\)-completely semiprime ideal of \(R\) and show that the converse does not hold. They also prove that if \(N_r(R)\) is a \(\sigma\)-completely semiprime ideal of \(R\), then \(N_r(R)={\mathcal N}(R)\) but the converse does not hold. However, \(N_r(R)={\mathcal N}(R)\) if and only if \(N_r(R)\) is a \(\sigma\)-completely semiprime ideal of \(R\) in the case when for each \(P\in\text{(m)Spect}_S(R)\), \(P\) is a \(\sigma\)-invariant ideal of \(R\). They note that there exists a ring with \(N_r(R)\neq{\mathcal N}(R)\) even though every strongly prime ideal of \(R\) is \(\sigma\)-invariant, while the condition `\(P\) is a \(\sigma\)-invariant ideal of \(R\) for each \(P\in\text{(m)Spect}_S(R)\)' cannot be replaced by the condition `\(N_r(R)\) is a \(\sigma\)-invariant ideal'. However, for a ring \(R\) the following conditions are equivalent: (1) \(N_r(R)\) is a \(\sigma\)-completely semiprime ideal of \(R\). (2) \(N_r(R)={\mathcal N}(R)\) and \(P\) is a \(\sigma\)-invariant ideal of \(R\) for each \(P\in\text{(m)Spect}_S(R)\). (3) \(P\) is a \(\sigma\)-ideal such that \(a\sigma(a)\in P\) implies \(a\in P\) for each \(P\in\text{(m)Spect}_S(R)\). This allows the authors to conclude that \(R\) is a \(\sigma\)-rigid ring if and only if \(R\) is a reduced ring, \(\sigma\) is a monomorphism and \(P\) is a \(\sigma\)-ideal for each \(P\in\text{(m)Spect}_S(R)\). Then they investigate the relationship between \(N_r(R[\![x;\sigma]\!])\) and \(N_r(R)\). They observe that if \(N_r(R)\) is a \(\sigma\)-completely semiprime ideal of \(R\), then \(N_r(R)[\![x;\sigma]\!]\) is an ideal of \(R[\![x;\sigma]\!]\). Let \(N_r(R)\) be a \(\sigma\)-completely semiprime ideal of \(R\) and \(p(x)=\sum_{i=0}^\infty a_ix^i\) and \(q(x)=\sum_{j=0}^\infty b_jx^j\) are elements of \(R[\![x;\sigma]\!]\). The authors show that \(p(x)q(x)\in N_r(R)[\![x;\sigma]\!]\) if and only if \(a_ib_j\in N_r(R)\) for all \(i\geqslant 0\) and \(j\geqslant 0\). Consequently, if \(N_r(R \) is a \(\sigma\)-completely semiprime ideal of \(R\), then \(N_r(R)[\![x;\sigma]\!]\) is a completely semiprime ideal of \(R[\![x;\sigma]\!]\). They also show that the condition `\(N_r(R)\) is a \(\sigma\)-completely semiprime ideal of \(R\)' is not superfluous. However, if \(N_r(R)\) is a \(\sigma\)-completely semiprime ideal of \(R\), then \(P[\![x;\sigma]\!]\) is a completely prime ideal of \(R[\![x;\sigma]\!]\) for each \(P\in\text{(m)Spect}_S(R)\). The authors also show that if \(N_r(R[\![x;\sigma]\!])={\mathcal N}(R[\![x;\sigma]\!])\), then \(N_r(R)={\mathcal N}(R)\) but the converse does not hold. Moreover, they show that in general, \(N_r(R[\![x;\sigma]\!])\neq{\mathcal N}(R[\![x;\sigma]\!])\) even if \(N_r(R)\) is a \(\sigma\)-completely semiprime ideal of \(R\). However, if \(N_r(R)\) is a \(\sigma\)-completely semiprime ideal of \(R\), then \(N_r(R[\![x;\sigma]\!])\) is a completely semiprime ideal of \(R[\![x;\sigma]\!]\) if and only if \(N_r(R)[\![x;\sigma]\!]=N_r(R[\![x;\sigma]\!])\) and that the condition `\(N_r(R)\) is a \(\sigma\)-completely semiprime ideal of \(R\)' is not superfluous. This result allows the author to conclude that if \(R\) is a \(\sigma\)-rigid ring, then \(N_r(R[\![x;\sigma]\!])\) is a completely semiprime ideal of \(R[\![x;\sigma]\!]\) if and only if \(N_r(R[\![x;\sigma]\!])=0\). The reader will notice few misprints in this paper but they are easy to correct.