Localization and Ideal Theory in Iterated Differential Operator Rings

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Abstract (in LaTeX)

We study primality, hypercentrality, simplicity, and localization and
the second layer condition in skew enveloping algebras and iterated
differential operator rings.  We give sufficient conditions for the
skew enveloping algebra of a nilpotent Lie algebra with coefficient
ring containing the rational numbers to be a simple ring, and we give
necessary and sufficient conditions in the case that the Lie algebra
is Abelian.  Our main results show that if $L$ is a finite dimensional
solvable Lie algebra and $R$ is an Artinian ring or a commutative
Noetherian algebra over $k$, then the skew enveloping algebra $R\#U(L)$
satisfies the second layer condition.  We discuss consequences of this
for localization and use the localization theory to state a classical
Krull dimension versus global dimension inequality when $k$ is
uncountable.

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