Matrix over a module

Let $𝑉$ be a (left) $𝑅$ -module. The matrix space $M 𝑚, 𝑛 (𝑉)$ is the (left) $𝑅$ -module #m/def/module

M 𝑚, 𝑛 (𝑉) = 𝑚 ⨁ 𝑖 = 1 𝑛 ⨁ 𝑗 = 1 𝑉

consisting of $𝑚 \times 𝑛$ rectangular arrays and acted on the matrix algebra $M ∙, 𝑚 (𝑅)$ so that for $𝐵 = (𝑏 𝑖 𝑗) 𝑚, 𝑛 𝑖 = 1, 𝑗 = 1 \in M 𝑚, 𝑛 (𝑉)$ and $𝐴 = (𝑎 𝑖 𝑗) ℓ, 𝑚 𝑖 = 1, 𝑗 = 1 \in M 𝑚, 𝑜 𝑝 𝑛 𝑀 𝑙 (𝑅)$ we have $𝐶 = 𝐴 𝐵 = (𝑐 𝑖 𝑗) ℓ, 𝑛 𝑖 = 1, 𝑗 = 1$

𝑐 𝑖 𝑗 = 𝑚 \sum 𝑘 = 1 𝑎 𝑖 𝑘 𝑏 𝑘 𝑗

In particular, $M 𝑚, 𝑛 (𝑉)$ is a (left) $M 𝑚, 𝑚 (𝑅)$ module.

Properties

Zero of a matrix over a ring

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