Left Module Does Not Necessarily Induce Right Module over Ring/Lemma
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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:
- $\quad \mathbf R \times \mathbf J = \begin{bmatrix} r_{1 1} \times j_1 + r_{2 1} \times j_2 & 0 \\ r_{1 2} \times j_1 + r_{2 2} \times j_2 & 0 \end{bmatrix} \in G$
$\blacksquare$