On sum annihilator ideals in Ore extensions

Abstract

A ring $R$ is called a left Ikeda-Nakayama ring (left IN-ring) if the right annihilator of the intersection of any two left ideals is the sum of the two right annihilators. As a generalization of left IN-rings, a ring $R$ is called a right SA-ring if the sum of right annihilators of two ideals is a right annihilator of an ideal of $R$. It would be interesting to find conditions under which an Ore extension $R[x; \alpha, \delta]$ is IN and SA. In this paper, we will present some necessary and sufficient conditions for the Ore extension $R[x;\alpha, \delta]$ to be left IN or right SA. In addition, for an $(\alpha,\delta)$-compatible ring $R$, it is shown that: (i) If $S = R[x;\alpha,\delta]$ is a left IN-ring with ${\rm{Idm}}(R) ={\rm{Idm}}(R[x;\alpha, \delta])$, then $R$ is left McCoy. (ii) Every reduced left IN-ring with finitely many minimal prime ideals is a semiprime left Goldie ring. (iii) If $R$ is a commutative principal ideal ring, then $R$ and $R[x]$ are IN. (iv) If $R$ is a reduced ring and $n$ is a positive integer, then $R$ is right SA if and only if $R[x]/(x^{n+1})$ is right SA.