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

Theorem

Let $\struct {S, +, \times}$ be a ring with unity.

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

Let:

$G = \set {\begin{bmatrix}

x & 0 \\ y & 0 \end{bmatrix} : x, y \in S }$

Then:

$G$ is a left ideal of $\struct {\map {\MM_S} 2, +, \times}$.

Proof

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

$(1): \quad G \ne \O$

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

$(3): \quad \forall \mathop{\mathbf J} \in G, \mathop {\mathbf R} \in \map {\MM_S} 2: \mathbf R \times \mathbf J \in G$

Condition $(1): \quad G \ne \O$

By definition of $G$:

$\quad \begin{bmatrix}

0 & 0 \\ 0 & 0 \end{bmatrix} \in G$

$\Box$

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

Let:

$\quad \mathbf X = \begin{bmatrix}

x_1 & 0 \\ x_2 & 0 \end{bmatrix}, \quad \mathbf Y = \begin{bmatrix} y_1 & 0 \\ y_2 & 0 \end{bmatrix} \in G$

Then:

$\quad \mathbf X - \mathbf Y = \begin{bmatrix}

x_1 - y_1 & 0 \\ x_2 - y_2 & 0 \end{bmatrix} \in G$

$\Box$

Condition $(3): \quad \forall \mathop{\mathbf J} \in G, \mathop{\mathbf R} \in \map {\MM_S} 2: \mathbf R \times \mathbf J \in G$

Let:

$\quad \mathbf J = \begin{bmatrix}

j_1 & 0 \\ j_2 & 0 \end{bmatrix} \in G, \quad \mathbf R = \begin{bmatrix} r_{1 1} & r_{2 1} \\ r_{1 2} & r_{2 2} \end{bmatrix} \in \map {\MM_S} 2$

Then:

$\quad \mathbf R \times \mathbf J = \begin{bmatrix}

r_{1 1} \times j_1 + r_{2 1} \times j_2 & 0 \\ r_{1 2} \times j_1 + r_{2 2} \times j_2 & 0 \end{bmatrix} \in G$

$\blacksquare$