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Question #18195

Show that for any finite-dimensional k-algebra R and any field extension K ⊇ k, (rad R)K ⊆ rad(RK).

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**1.**For a finite-dimensional k-algebra R, let T(R) = rad R + [R,R], where [R,R] denotes the subgroup of**2.**Let R be a finite-dimensional k-algebra which splits over k. Show that any k-subalgebra C ⊆**3.**Let R be a left artinian ring and C be a subring in the center Z(R) of R. If R is a finite-dimension**4.**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**5.**Let R be a finite-dimensional k-algebra, M be an R-module and E = EndRM. Show that if f ∈ E**6.**Let R be a finite-dimensional k-algebra which splits over k. Show that, for any field K ⊇ k**7.**Let M,N be finite-dimensional modules over a finite-dimensional k-algebra R. For any field K &su

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#239218 on Sep 2019

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