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.The authors retain all rights to the original work without any restrictions.
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