Nilradicals of skew Hurwitz series of rings

Abstract

‎For a ring endomorphism α of a ring R, ‎Krempa called α a rigid endomorphism if aα(a)=0 implies a = 0 for a in R. ‎A ring R is called rigid if there exists a rigid endomorphism of R. ‎In this paper‎, ‎we extend the α-rigid property of a ring R to the upper nilradical N_r(R) of R. ‎For an endomorphism α and the upper nilradical N_r(R) of a ring R, ‎we introduce the condition (*): ‎N_r(R) is a α-ideal of R and aα(a) in N_r(R) implies a in N_r(R) for a in R. ‎We study characterizations of a ring R with an endomorphism α satisfying the condition (*), ‎and we investigate their related properties‎. ‎The connections between the upper nilradical of R and the upper nilradical of the skew Hurwitz series ring (HR,α) of R are also investigated‎.

Keywords

• Skew Hurwitz series
• Wedderburn radical
• nil radical