Bulletin of the
Korean Mathematical Society BKMS

For a ring endomorphism $\sigma$ of a ring $R$, J. Krempa called $\sigma$ a rigid endomorphism if $a\sigma(a)=0$ 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 $\sigma$-rigid property of a ring $R$ to the upper nilradical $N_r(R)$ of $R$. For an endomorphism $\sigma$ and the upper nilradical $N_r(R)$ of a ring $R$, we introduce 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 $a\in R$. We study characterizations of a ring $R$ with an endomorphism $\sigma$ satisfying the condition $(*)$, and we investigate their related properties. The connections between the upper nilradical of $R$ and the upper nilradical of the skew power series ring $R[[x; \sigma]]$ of $R$ are also investigated.

Keywords: rigid endomorphisms, the upper nilradicals, skew power series rings

MSC numbers: 16W20, 16N40, 16W60