Module theory MOC

Matrix algebra over a ring

Let 𝑅 be a ring. The matrix algebra M∙,∙⁡(𝑅) over 𝑅 is a free 𝑅-bimodule with decomposition module

M∙,∙⁡(𝑅)=∞⨁𝑚=1∞⨁𝑛=1M𝑚,𝑛⁡(𝑅)

where

M𝑚,𝑛⁡(𝑅)=𝑚⨁𝑖=1𝑛⨁𝑗=1𝑅

is an 𝑅-bimodule consisting of 𝑚 ×𝑛 rectangular arrays with entries in 𝑅 and addition and scalar multiplication defined pointwise. Given 𝐴 =(𝑎𝑖𝑗)ℓ,𝑚𝑖=1,𝑗=1 ∈Mℓ,𝑚⁡(𝑅) and 𝐵 =(𝑏𝑖𝑗)𝑚,𝑛𝑖=1,𝑗=1 ∈M𝑚,𝑛⁡(𝑅) we define the matrix product 𝐶 =𝐴𝐵 =(𝑐𝑖𝑗)ℓ,𝑛𝑖=1,𝑗=1

𝑐𝑖𝑗=𝑚∑𝑘=1𝑎𝑖𝑘𝑏𝑘𝑗

which may be extended to the whole of M∙,∙⁡(𝑅) by defining M𝑘,ℓ⁡(𝑅)M𝑚,𝑛⁡(𝑅) =0 for ℓ ≠𝑚.

Further operations

• Matrix determinant
• Matrix trace
• Matrix transpose

Special cases

• Matrix algebra over a field
• Matrix over a module

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