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Question #18102

char(k) = 3, and let G = S3. Compute the Jacobson radical J = rad(kG), and the factor ring kG/J.

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

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