Let \(R\) be an associative ring. A ring \(R\) is said to be right APIP if the right annihilator of any nonzero element \(a\) in \(R\) contains the principal ideal of \(R\) generated by some power of any right zero-divisor of \(a\), equivalently, \(ab = 0\) for \(a,b\in R\) implies \(Rb^mR\subseteq r_R(a) \) (i.e., \(aRb^m=0\)) for some \(m\geq 1\). The left APIP can be defined by symmetry.
In this manuscript, the authors investigate the structure of rings with two conditions. First, the right annihilator of some power of any element is an ideal. Second, the right annihilator of any nonzero element \(a\) contains an ideal generated by some power of any right zero-divisor of the element \(a\). Additionally they establish that these conditions are shown to be not right-left symmetric. For a prime two-sided APIP ring \(R\), the authors 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. Also, they provide a variety of interesting examples that describe the classes of rings related to these properties.