Localization and Ideal Theory in Noetherian Strongly Group-graded Rings

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Abstract (in LaTeX)

We study primality, hypercentrality, simplicity, and localization and
the second layer condition in skew group rings and group-graded rings.
We give necessary and sufficient conditions for the skew group ring of
a torsion-free nilpotent group to be a simple ring, and if the coefficient
ring is commutative, we give necessary and sufficient conditions for the
skew group ring of an Abelian group to be simple.  Our method involves
showing certain group-graded rings are hypercentral.  Our main results
show that if $G$ is a polycyclic-by-finite group and $R$ is an Artinian
ring or a commutative Noetherian ring, then a strongly $G$-graded ring
with base ring $R$ satisfies the second layer condition.  We discuss
consequences of this for localization in such rings.

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