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We discuss the condition that every nonzero right annihilator of an element contains a nonzero ideal, as a generalization of the insertion-of-factors-property. A ring with such condition is called {\it right AP}. We prove that a ring $R$ is right AP if and only if $D_n(R)$ is right AP for every $n\geq 2$, where $D_n(R)$ is the ring of $n$ by $n$ upper triangular matrices over $R$ whose diagonals are equal. Properties of right AP rings are investigated in relation to nilradicals, prime factor rings and minimal order.
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