Solution: Let $R$ as the set of all $2\times 2$ upper triangular matrices over $\mathbb{R}$ (we can take any non-zero field). \[ R = \left\{ \begin{pmatrix} a & b \\ 0 & c \\ \end{pmatrix} : a,b,c\in \mathbb{R} \right\}. \] Let \[ x = \begin{pmatrix} 1 & 0 \\ 0 & 0 \\ \end{pmatrix}. \]

Note that \begin{align*} Rx & = \left\{ r \begin{pmatrix} 1 & 0 \\ 0 & 0 \\ \end{pmatrix} : r\in R \right\} \\[1ex] & = \left\{ \begin{pmatrix} a & b \\ 0 & c \\ \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & 0 \\ \end{pmatrix} : a,b,c \in \mathbb{R} \right\} \\[1ex] & = \left\{ \begin{pmatrix} a & 0 \\ 0 & 0 \\ \end{pmatrix}: a\in \mathbb{R} \right\} . \end{align*}

Similarly \begin{align*} xR & = \left\{ \begin{pmatrix} 1 & 0 \\ 0 & 0 \\ \end{pmatrix}r : r\in R \right\} \\[1ex] & = \left\{ \begin{pmatrix} 1 & 0 \\ 0 & 0 \\ \end{pmatrix} \begin{pmatrix} a & b \\ 0 & c \\ \end{pmatrix}: a,b,c\in \mathbb{R} \right\} \\[1ex] & = \left\{ \begin{pmatrix} a & b \\ 0 & 0 \\ \end{pmatrix} \right\} . \end{align*} It is clear that $Rx \subsetneqq xR$.