In a noncommutative ring (with identity), is it possible for an element which does not possess left or right inverses to generate the entire ring? i.e. $(r)=R$, where (r) is the twosided ideal generated by $r\in R$ ?
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Yes, for example $r=\begin{pmatrix}1&0\cr0&0\end{pmatrix}$ generates the whole 2x2 matrix ring, $$ \begin{pmatrix}1&0\cr0&0\end{pmatrix} + \begin{pmatrix}0&0\cr1&0\end{pmatrix} \cdot \begin{pmatrix}1&0\cr0&0\end{pmatrix} \cdot \begin{pmatrix}0&1\cr0&0\end{pmatrix} = \begin{pmatrix}1&0\cr0&1\end{pmatrix}. $$ 


Take for example the endomorphism ring of a finite dimensional vector space. There are only trivial two sided ideals. So the ideal generated by a non zero element is the whole ring. 

