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Question #18101

Let k be a commutative ring and G be any group. If kG is left artinian, show that kG is right artinian.

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**1.**Let k be a commutative ring and G be any group. If kG is left noetherian, show that kG is right noet**2.**Let A be a normal elementary p-subgroup of a finite group G such that the index of the centralizer C**3.**Let R be a commutative domain that is not a field. Show that not always R is not J-semisimple implie**4.**Let R be a commutative domain that is not a field. Show that if R is not J-semisimple then R is semi**5.**Let R be a commutative domain that is not a field. If R is semilocal, show that R is not J-semisimpl**6.**For any ring R with Jacobson radical J, we have an exact sequence of groups (∗) 1&rar**7.**If J is a nil ideal and the ring R/J has prime characteristic p, show that 1 + J is a p-group.

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