Ring of Square Matrices over Commutative Ring with Unity

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Theorem

Let $R$ be a commutative ring with unity.

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $\struct {\map {\MM_R} n, +, \times}$ denote the ring of square matrices of order $n$ over $R$.


Then $\struct {\map {\MM_R} n, +, \times}$ is a ring with unity.

However, for $n \ge 2$, $\struct {\map {\MM_R} n, +, \times}$ is not a commutative ring.


Proof

From Ring of Square Matrices over Ring with Unity we have that $\struct {\map {\MM_R} n, +, \times}$ is a ring with unity.

However, Matrix Multiplication is not Commutative.

Hence $\struct {\map {\MM_R} n, +, \times}$ is not a commutative ring for $n \ge 2$.

For $n = 1$ we have that:

\(\, \displaystyle \forall \mathbf A, \mathbf B \in \map {\MM_R} 1: \, \) \(\displaystyle \mathbf A \mathbf B\) \(=\) \(\displaystyle a_{11} b_{11}\) where $\mathbf A = \begin {pmatrix} a_11 \end {pmatrix}$ and $\mathbf B = \begin {pmatrix} b_11 \end {pmatrix}$
\(\displaystyle \) \(=\) \(\displaystyle b_{11} a_{11}\) as $R$ is a commutative ring
\(\displaystyle \) \(=\) \(\displaystyle \mathbf {B A}\)

Thus, for $n = 1$, $\struct {\map {\MM_R} n, +, \times}$ is a commutative ring.

$\blacksquare$


Notation

When referring to the operation of matrix multiplication in the context of the ring of square matrices:

$\struct {\map {\MM_R} n, +, \times}$

we must have some symbol to represent it, and $\times$ does as well as any.

However, we do not use $\mathbf A \times \mathbf B$ for matrix multiplication $\mathbf A \mathbf B$, as it is understood to mean the vector cross product, which is something completely different.


Sources