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Question #16899

If RV is a simple module over a commutative ring R, show that any direct product of copies of V is semisimple.

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**1.**Let U, V be semisimple modules over a commutative ring. Is HomR(U, V ) also semisimple?**2.**Let U, V be modules over a commutative ring R. If U or V is semisimple, show that U ⊗R V**3.**Show that for a semisimple module M over any ring R, the following conditions are equivalent: (1) M**4.**Let R be a right semisimple ring. For x, y ∈ R, show that Rx =Ry iff x = uy for some unit u**5.**Let R be a (left) semisimple ring. Show that, for any right ideal I and any left ideal J in R, IJ =**6.**Let R be the (commutative) ring of all real-valued continuous functions on [0, 1]. Is R a semisimple**7.**What are the semisimple Z-modules? (Characterize them in terms of their structure as abelian groups.

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