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Question #18198

If K ⊇ k is a splitting field for a finite-dimensional k-algebra R, does it follow that K is also a splitting field for any quotient algebra R of R?

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**1.**Let R be a finite-dimensional k-algebra and let L ⊇ K ⊇ k be fields. Assume that L**2.**Show that for any finite-dimensional k-algebra R and any field extension K ⊇ k, (rad R)K &a**3.**For a finite-dimensional k-algebra R, let T(R) = rad R + [R,R], where [R,R] denotes the subgroup of**4.**Let R be a finite-dimensional k-algebra which splits over k. Show that any k-subalgebra C ⊆**5.**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**6.**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**7.**Let R be a finite-dimensional k-algebra, M be an R-module and E = EndRM. Show that if f ∈ E

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