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Question #18191

Let R be a left artinian ring and C be a subring in the center Z(R) of R. If R is a finite-dimensional algebra over a subfield k ⊆ C, show that rad C = C ∩ rad R.

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**1.**Let R be a left artinian ring and C be a subring in the center Z(R) of R. Show that Nil C = C &c**2.**Let R be a finite-dimensional k-algebra, M be an R-module and E = EndRM. Show that if f ∈ E**3.**Let R be a finite-dimensional k-algebra which splits over k. Show that, for any field K ⊇ k**4.**Let M,N be finite-dimensional modules over a finite-dimensional k-algebra R. For any field K &su**5.**For any nonzero ring k and any group G, show that the group ring kG is von Neumann regular iff k is**6.**Show that statement "for any von Neumann regular ring k, any finitely generated submodule M**7.**For any von Neumann regular ring k, show that any finitely generated submodule M of a projective k-m

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