Let R be a ring n belongs to N and MnR denote the ring of al

Let R be a ring, n belongs to N and M_n(R) denote the ring of all n times n matrices over R. Prove that (M_n(R))[x] M_n(R[x]) (M_n(R))[x] M_n(R[x]).

Solution

a) Let R be a ring and S = Mn(R). Then S is finitely generated as an R-module, generated by n2
elements. If R is Artinian or Noetherian, then by previous result, so is S as an R-module. Now any left
ideal of S is also a left R-module. Hence every chain of left ideals is a chain of left R-submodules of S.
Thus if R is Artinian (Noetherian) then S is also Artinian (Noetherian) ring.

b)Let R be a ring and M = h(x1, x2, · · · , xni) be a finitely generated R-module.
Let : Rn ! M be defined by (r1, r2, · · · , rn) =summation

i=1
rixi. Then it is easy to see that is an onto
R-module homomorphism. Thus Rn/ker = M.
Now if R is Artinian or Noetherian, so is Rn/ker. Hence so is M.