Nilradicals of skew power series rings

Abstract

For a ring endomorphism sigma of a ring R, J. Krempa calledsigma a rigidendomorphism if asigma(a)=0 implies a=0 for ain R. Aring R is called rigid if there exists a rigidendomorphism of R. In this paper, we extend the sigma-rigid property of a ringR to the upper nilradical N_r(R) of R. For an endomorphismsigma and the upper nilradical N_r(R) of a ring R, weintroduce the condition (*): N_r(R) is a sigma-ideal ofR and asigma(a)in N_r(R) implies ain N_r(R) for ainR. We study characterizations of a ring R with an endomorphismsigma satisfying the condition (*), and we investigate theirrelated properties. The connections between the upper nilradicalof R and the upper nilradical of the skew power series ringR[[x; sigma]] of R are also investigated.