It is well-known that a group ring of a finite group is semi-simple, and since profinite-groups are projective limits of finite groups, I am thinking per chance profinite groups still possess some good properties inherited from the finite components. As I have found thus far no reference in the direction, I ask here the question: Is the group ring of a profinite group (necessarily) (semi)-hereditary?
Thanks in advance.
Is the group ring of a pro-finite group (semi)-hereditary?
2
$\begingroup$
abstract-algebra
group-theory
ring-theory
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0The group ring over what field (or ring)? – 2012-12-09
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0Basically over the ring of integers. Thanks for the attention. – 2012-12-09
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0The group ring of a finite group over the integers is not semi-simple. Indeed, any module $M$ has proper submodules $nM$ for any $n\in \mathbb{N}$. So there are no simple modules. – 2012-12-09
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0I know that at least it is semi-simple over $C$. So let us suppose that it is semi-simple over some field $k$, and then ask if the profinite group is (semi)-heriditary? – 2012-12-11