A.D. Bell: Abstract

## When are all prime ideals in an Ore extension Goldie?

Allen D. Bell

### Publication Information

Appeared in Communications in Algebra 13#8 (1985), pp. 1743-1762
Math Reviews: 86j:16003

### Abstract (in LaTeX)

We show that if $\phi$ is an automorphism of $R$, then all
prime ideals of the skew polynomial ring $R[x;\phi]$ are
right Goldie if and only if (a) every prime ideal of $R$ is
right Goldie and (b) every strongly $\phi$-prime ideal of
$R$ is a finite intersection of prime ideals.  [We say $I$
is strongly $\phi$-prime if $\phi(I)=I$ and if whenever
$\phi^n(J)K$ is contained in $I$ for ideals $J,K$ and all
large $n$, we have either $J$ or $K$ contained in $I$.]
We obtain a similar result for skew Laurent rings, and
we obtain some partial results for skew polynomial rings
with both automorphism and derivation.

We show that if $R$ has the a.c.c. on ideals, then all
prime ideals in $R[x;\phi]$ are right Goldie if and only
if all prime ideals in $R$ are right Goldie, but we give
an example showing that this is not true in general.  We
give some other negative examples.