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ABSTRACT
In this paper, we extend the recent work of C. FaithCitation[1] in which it is proved, using results of R.C. Shock,Citation[2] that over a commutative ring , the associated primes of the polynomial ring
(viewed as a module over itself) are all extended; that is, every
may be expressed as
, where
. This result was originally proved in 1974 by Brewer and Heinzer using localization theory (seeCitation[3]), but our work in this paper more closely parallels that of Faith. We will show that the result still holds in the more general setting of a polynomial module
over a skew polynomial ring
, with possibly noncommutative base ring
, provided a natural
-compatibility assumption on
(as defined in Sec. 2 below) is fulfilled. Moreover, if we assume that
, we can make the set of Laurent polynomials
into a module over
and
, and the main result still holds in these cases as well (for
-compatible
). Finally, we also show that the result does not extend to the case of skew formal power series rings and skew Laurent series rings in general, although it does if
is a left perfect ring.
ACKNOWLEDGMENTS
Much of this paper was completed while I resided with my parents, Arthur and Juliann Annin, and they are deeply thanked for their gracious hospitality and endless encouragement and support. Dr. T.Y. Lam and Dr. Greg Marks also deserve special thanks for reading preliminary versions of this paper and making numerous thoughtful comments and suggestions on this work; their guidance has been invaluable. Finally, I sincerely thank Dr. Will Murray for many helpful discussions on the examples that appear in this paper.
Notes
Recall that an ideal in a (possibly noncommutative) ring
is said to be a prime ideal if
and for
,
implies that
or
. Other characterizations and much more information regarding prime ideals in the noncommutative setting can be found in Chapter 4 of Citation[5].
The ring consists of all Laurent series of the form
with
and coefficients
, subject to the usual twisted multiplication rule, which now requires
Here, means
The reader can check that if we instead take in this example, with the action of
on
given by
, then
is
-compatible in this case (for the same
). This provides an example in which Theorem 2.2 may be applied for a choice of
which is
an automorphism of
; the conclusion is that
.
Indeed, this assumption is needed in order to define the multiplicative structure on the ring .
This module consists of all polynomials of the form , where
and
.
Alternatively, can be seen to be prime in
from the number-theoretic result that
is isomorphic to the ring of
-adic integers.
The element used to define the embedding
in the proof of Theorem 2.2 may be replaced here by any nonzero coefficient of
.
Semiperfect rings are discussed at length in Chapter 8 ofCitation[5]. They are the semilocal rings such that idempotents of can be lifted to
. This class includes all local rings and all left (resp. right) perfect rings. since local rings are always semiperfect.
We mention briefly an alternative approach. This approach, while applicable to the Proposition 5.1 example as well, is nonconstructive in nature and provides less explicit information, so in the earlier case we preferred the direct computation. Nevertheless, we know that Ass since the ring
is Noetherian. But every
must contain
, since
annihilates
. This implies that
cannot be of the form
for
.
One easily uses our earlier conclusion that is an irreducible element of the unique factorization domain
to show that
is also an irreducible element of the unique factorization domain
.