Find a right Artinian right principal ideal ring $R$ with a right ideal $I$ such that $I$ is not equal to right annihilator of an element of $R$. Any comment, if any, is wellcome. Thank you very much.

An element $s$ of a group $G$ is a logical generator of $G$ iff every element of $G$ can be defined in the first order language of groups with $s$ as a parameter. In this case we may call $G$ a logically cyclic group. This is equivalent to say that for any elementary extension $G^{\ast}$ of $G$ ...