Abstract Algebra Answers

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

Let k ⊆ K and G be as above. Show that a kG-module M is semisimple iff the KG-module MK = M ⊗k K is semisimple.

Let k ⊆ K be two fields and G be a finite group. Show that rad (KG) = (rad kG) ⊗k K.

Show that if k0 is any finite field and G is any finite group, then (k0G/rad k0G) ⊗k0 K is semisimple for any field extension K ⊇ k0.

Let H be a normal subgroup of G. If H is finite, show that I is also nilpotent.

Let H be a normal subgroup of G. If rad kH is nilpotent, show that I is also nilpotent.

Let H be a normal subgroup of G. Show that I = kG • rad kH is an ideal of kG.

If k is an uncountable field, show that, for any group G, rad kG is a nil ideal.

Deduce that, if kH is J-semisimple for any finitely generated subgroup H of G, then kG itself is J-semisimple.

For any subgroup H of a group G, show that kH ∩ U(kG) ⊆ U(kH) and kH ∩ rad kG ⊆ rad kH.