Find a ring with nonzero elements $a,b,c$ such that $ab=ca=0$ but $ba\neq 0$.

Question

Is there any element $a$ in a ring $R$ with identity (non commutative) such that $aR = Ra$ with non zero elements $b$ and $c$ in $R$ having the property that $ab = 0$, $ba\ne 0$, but $ca = 0$?

Answer 1

There is an example (that is apparently what you're looking for) of a ring with $8$ elements satisfying this. Take $R$ to be the upper triangular matrices over the field $F_2$ of two elements.

Let $a=\begin{bmatrix}0&1\\0&0\end{bmatrix}$.

You have that $aR=Ra=\begin{bmatrix}0&F_2\\0&0\end{bmatrix}$.

Let $b=\begin{bmatrix}1&1\\0&0\end{bmatrix}$.

Then you have $ab=0$ and $ba=a\neq 0$.

For $c$ you can use $c=\begin{bmatrix}0&1\\0&1\end{bmatrix}$.

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A ring in which $ab=0$ implies $ba=0$ for all elements $a,b$ is called a reversible ring.