Right Module Does Not Necessarily Induce Left Module over Ring/Lemma

Theorem
Let $\struct {S, +_S, \times_S}$ be a ring with unity

Let $0_S, 1_S$ be the zero and unity of $S$ respectively.

Let $\struct {\map {\mathcal M_S} 2, +, \times}$ denote the ring of square matrices of order $2$ over $S$.

Let $G = \set {\begin{bmatrix} x & y \\ 0_S & 0_S \end{bmatrix} : x, y \in S }$

Then:
 * $G$ is a right ideal of $\struct {\map {\mathcal M_S} 2, +, \times}$.