A Reduced Non-McCoy Ring with the Annihilator Condition and $(A_2)$

Abstract

It has been asked whether a reduced non-McCoy ring can satisfy both the annihilator condition and $(A_n)$ for some $n\ge 2$. In this work, we prove that such rings exist exactly when $n=2$, by constructing a countable example and showing that the annihilator condition together with $(A_3)$ forces a ring to be McCoy.