Matrix algebra over a ring

Let $𝑅$ be a ring. The matrix algebra $M ∙, ∙ (𝑅)$ over $𝑅$ is a free $𝑅$ -bimodule with decomposition #m/def/module

M ∙, ∙ (𝑅) = \infty ⨁ 𝑚 = 1 \infty ⨁ 𝑛 = 1 M 𝑚, 𝑛 (𝑅)

where

M 𝑚, 𝑛 (𝑅) = 𝑚 ⨁ 𝑖 = 1 𝑛 ⨁ 𝑗 = 1 𝑅

is an $𝑅$ -bimodule consisting of $𝑚 \times 𝑛$ rectangular arrays with entries in $𝑅$ and addition and scalar multiplication defined pointwise. Given $𝐴 = (𝑎 𝑖 𝑗) ℓ, 𝑚 𝑖 = 1, 𝑗 = 1 \in M ℓ, 𝑚 (𝑅)$ and $𝐵 = (𝑏 𝑖 𝑗) 𝑚, 𝑛 𝑖 = 1, 𝑗 = 1 \in M 𝑚, 𝑛 (𝑅)$ we define the matrix product $𝐶 = 𝐴 𝐵 = (𝑐 𝑖 𝑗) ℓ, 𝑛 𝑖 = 1, 𝑗 = 1$

𝑐 𝑖 𝑗 = 𝑚 \sum 𝑘 = 1 𝑎 𝑖 𝑘 𝑏 𝑘 𝑗

which may be extended to the whole of $M ∙, ∙ (𝑅)$ by defining $M 𝑘, ℓ (𝑅) M 𝑚, 𝑛 (𝑅) = 0$ for $ℓ \neq 𝑚$ .

Further operations

Special cases

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