Left Module Does Not Necessarily Induce Right 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 $G = \set {\begin{bmatrix} x & 0_S \\ y & 0_S \end{bmatrix} : x, y \in S }$

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

Proof
From Test for Left Ideal, the following need to be proved:


 * $(1): \quad G \ne \varnothing$


 * $(2): \quad \forall \mathop {\mathbf X}, \mathop{\mathbf Y} \in G: \mathbf X + \paren {-\mathbf Y} \in G$


 * $(3): \quad \forall \mathop{\mathbf J} \in G, \mathop{\mathbf R} \in \map {\mathcal M_S} 2: \mathbf R \circ \mathbf J \in G$

Condition $(1): \quad G \ne \varnothing$
By definition of $G$, $\begin{bmatrix} 0_S & 0_S \\ 0_S & 0_S \end{bmatrix} \in G$