Summary: A ring \(R\) has right property \((A)\) whenever a finitely generated two-sided ideal of \(R\) consisting entirely of left zero-divisors has a non-zero right annihilator. As the main result of this paper we give answers to two questions related to property \((A)\), raised by C. Y. Hong et al. [J. Algebra 315, No. 2, 612–628 (2007; Zbl 1156.16001)]. One of the questions has a positive answer and we obtain it as a simple conclusion of the fact that if \(R\) is a right duo ring and \(M\) is a u.p.-monoid (unique product monoid), then \(R\) is right \(M\)-McCoy and the monoid ring \(R[M]\) has right property \((A)\). The second question has a negative answer and we demonstrate this by constructing a suitable example.