We study the following two conditions in rings: (i) the right annihilator of some power of any element is an ideal, and (ii) the right annihilator of any nonzero element $a$ contains an ideal generated by some power of any right zero-divisor of the element $a$. We investigate the structure of rings in relation to these conditions; especially, a ring with the condition (ii) is called right APIP. These conditions are shown to be not right-left symmetric. For a prime two-sided APIP ring $R$ we prove that every element of $R$ is either nilpotent or regular, and that if $R$ is of bounded index of nilpotency then $R$ is a domain. We also provide several interesting examples which delimit the classes of rings related to these properties.
BAECK, JONGWOOK; KIM, NAM KYUN; KWAK, TAI KEUN; and LEE, YANG
"Structure of annihilators of powers,"
Turkish Journal of Mathematics: Vol. 46:
5, Article 22.
Available at: https://journals.tubitak.gov.tr/math/vol46/iss5/22