Example

Let $a$ be an element of A which is neither invertible nor a right zero divisor. Then $\mathrm{dp}_A(A/Aa)=1$.

Indeed, by the exact sequence $0\longrightarrow A_s\xrightarrow{\varphi} A_s\longrightarrow A/Aa\longrightarrow0$, where $\varphi(x)=xa$, we have $\mathrm{dp}_A(A/Aa)\leq1$. If $\mathrm{dp}_A(A/Aa)<1$, then $A/Aa$ is projective, and there exists an a-linear mapping $\psi:A_s\longrightarrow A_s$ such that $\psi\circ\varphi=\mathrm{Id}$; this implies

and $a$ is invertible.