J(Mat n R Mat n J(R) Here is an outline of a proof (a) If A is a left R module, consider the elements of A n A A A (n summands) as column vectors then A n is a left (Mat n R) module (under ordinary matrix multiplication) (b) If A is a simple R module, A n is a simple (Mat n R) module (c) J(Mat n R)...

To prove that JMatnR MatnJR we need to show both inclusions JMatnR MatnJR This means every element of JMatnR is also an element of MatnJR MatnJR JMatnR This means every element of MatnJR is also an el ...

J(Mat n R} = Mat n J(R). Here is an outline of a proof: (a) If A

Question:

J(Mat_nR} = Mat_nJ(R). Here is an outline of a proof:

(a) If A is a left R-module, consider the elements of Aⁿ = A ⊕ A ⊕· · · ⊕ A (n summands) as column vectors; then Aⁿ is a left (Mat_nR)-module (under ordinary matrix multiplication).

(b) If A is a simple R-module, Aⁿ is a simple (Mat_nR)-module.

(d) Mat_nJ(R) ⊂ J(Mat_nR).

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