On Goldie Prime Ideals of Ore Extensions

Abstract

Let R be a ring and α an injective endomorphism of R, which is not assumed to be surjective. Necessary and sufficient conditions are given for all prime ideals in a skew polynomial ring R[x; α] or skew Laurent ring R[x, x −1; α] to be left Goldie. As a consequence, we obtain a generalization of a result of Goldie and Michler.

Full text

PRIME IDEALS OF ORE EXTENSIONS
ANDRE LEROY
UNTVERSITY
oF VALnNCIENNES
LE MONT HOUY
59326 VALENcInNNES
FnANcE.
JERZY
MATCZUK
UNIVERSITY
oF WARSA\Y
00-901 WARSAW, PI(IN
PoLAND.
Dedi.â.ie.l
to the mempry of Prôfèssôr R.hert R lva.field
Abstrrcr. For the Ore extension RIt,S,rl, where -R
is a prime dng, we
describe
prime ideals having
,ero intersection
with .R-
Introduction. The structure of prime ideals of various kinds of ring ex-
.ensiors has bcen invesrigatcd during the last ferv years. Normalizing ex-
tensions
([11]), crossed
products
([2],110]), enveloping rings
(11r],[12]) arrd
Or.c
r:xrensions
(l1l,l3l,i.li,t5l.16l.l7l)
were,
in pariicular. studied.
In iî], 16]
prirnes
of Ore extensions
over commutative
noetheriaû rilrqs
',vcre
corsiCereo.
In 121, f;l an<l
{12],
prime ideels.
disjoinr from rhc 'o.fr
cient r;rg, of Orc extensions o{ derivarion ["vpe
were described The casc
of Orc extensions of auiomorphism type has been deait receotlf in {1]. l*1.
The aim of this paper is to studl' these prime ideals of Ore extensiots.
rvhich have zero intersection vith coemcient ring. The methods we 'rse
irre
based
on l9l.
Throughout the paper -R
rviil denote a prime ring rvhile ? rvill srand lor
the s-vnmetric quotient ritg of R. Recall that the left Martitdale quoiiett
rioe of R isdefin'das OtrqJ= lim Hornrr.pl
,pR,, vhêre-F
is
'h'fi1-er
' -t€r
of aù non-zero ideals ol .R, anci
? can be considered as a subring of Ç(Ê)
consisting of such eiemenrs q € Q(E) thar 9I c Ê ior some ideai 1 € .F
depending
on g.
Ttre paper wæ rvliLLen while ihe auihoB were staying ai Instiiùt d'Esiudi6 Caialans,
Centre de Rece.ca Matematica, Spain.
Typesei b], -,tfis-TErx

5 and D will stand for arr automorphism and S-derivation of Ê, respec-
tively. Reca.ll
ihat at,9-derivation D is a-n endomorphism of the addiiive
group of R such that
D(aù) : D(a)ù + s(a)D(t) for all a,
ù
E R
In case
5 is ihe identiiy, D is a:r ordinary derivation. For each c ê -R
we
'will denoie by D",5 the S derivation of Ê deûned
bv D.,5(a) : ce - S(a)c
for all a € R. The Ore extension Ê!,5, D] is the ring of poLynomials in t
over E, with multiplication determined by the rule
ia : S(o)1
+ D(a) for all a in R
For /(1) e nl1,5, rl, deg
f(t) will denote
the degree
of the polvnomial
I(1).
It is rvell kûown that both 5 and D have unique extensions to T There
fore we can consider
the over
riûg ?1i,5'rl of Plf,S,D]. trVe
will give
a complete
description
of ?-disjoint prime ideals of ?li,5,D]. Next rve
will present a one Lo-one corfèspondence
between ? disjoint primes of
?!, 5, D] and Ë-disjoint prirres of E!,5, D], provided one of the follorving
conditions is satisfied :
a.) .R is symnetrically closed
(i e. ? = .R)
b) R is left and right nceiherian
c) E satisfres
the descending
chain condition on lruo-sided ideals
d) 5 and D commule and another minor iechnicaL assumptioû (cJ'
Prop.2.9.)
The above results lead io a. full description of .E disjoint prime ideals of
Rfl,5, Dl in these
câses.
Invariaut polynomials. RecalL
th:rt if /(1) € Ê11,5,
Dl is a rnonic
poi5'
nomial. lhen /(i) is invariant if
(i) /(1)l = (1
+ d)l(t) for sorne
.} € E.
and
(ii) for any r e R l(,1)r = s"(')/(1), rvhere
n = dee
l(1).
ff /(1) € Ê[1,5, D] is monic invariant then clearLy
ihe left ideal generared
by l(r) is trvo sided
and moreover
Rl1'S'D)l?) - /(1)RF,5,Dl. Con'
versely, as the foilorving lemma shows,
it is possible to associate
in a unique
rvay a monic invâriant polynomial to arrv non-zero ideal of Ê[i,5, D]
Lemma 1.1. (Prop.2.1, Cot 2.2
[9])
(1) For a.ny
non-zeto idea)
I of R{.1.5,D) rl.bere exists a unique motjc in-
variant polynomial lr(1) €Tll,S,D) having the fdlowing propertjes:
(i) dee
lr(r) - ntn{,les s(1)l0 + s(1) e I} = t and every
polvnomiat
9(1)
e I of degree
n is o[ the form g(1): '111(1)
for some
a € R'
(ii)
I c ?[r, s,
D]lI(1) . nli,
s, Dl.

(2) If I ;s an ideal of Tll,S,Dl then the polynomial f1() defined in (1)
belongs to Tl1,S,Dl. I
The polynomial fr(l) from the above lemma will be ca.lled
the irvârianr
polynomia.l associated to I.
In the sequel we wiil need the following simple observation.
Lemma 1.2. Let I C I be non-zero ideaJs
of Rll,S,Dl. ?àen tàere is a
monic invar|.ar't
poilnor!.iai
À(1)
€al11,S,Dl sûcÀ t'Àar
/r(t) : h(1)f
r(J).
Proof, By Lemma 1.1, there is a non zero idea-l,{
of -Rsuchthat
//r(i) C
I. Since
I C "I, Lemma 1.1
yieids
that for aûy a € ,4 there
is 9.('l) €
1'!,5, Dl such ihat e
fr(1)
= sù(1) lr(1)
f-'(1) is a monic polynomial, thus rve can divide lr(1) on the right oy
l7(1) gettiog fr(l) = À(t)ly(t) * r(l) lor some À(i)Jr(l) € ?{1,.9,D1 rvith
des r(t) a des
f 1(1).
Therefore, for any a € â we have
s"(t)f1(t): cf1(r)
: cÀ(r)17(r)+
cr(r)
and, consequently, (9"(1)
- aÀ(l))lr({) = ar(1). Comparing degrees
of
poiynornials appearing on both sides in the above
equality we get ar(l) :0
for all c € A. This implies
r(t) : 0 and |r(t): nl)t'1(t). Now one can
easily check that À(1) is a moric invariant poLynomial. I
Let M(i) € T[], 5, Dl denote a mo c inva-rianL poiynomial
of minima]
non-zero
degree.
provided
such
poivnomial
exisrs: otherrvise
M(l) = 1.
In order to describe
prime
ideals in R{l,5, Dl rve rvill
need a <lescription
of bhe certer Z of1:lt,S,D) and sone properties
of invariant polynomials.
C5.p rvill denote the ring of al1 certrâl elements
in T which are 5 and D
tnvanan[.
Proposition 1.3. t) (Th. 3.6 and 3.7
19])
Tlere exr'st an irvenible I € ?
andl ] 0 sucÀ that Z =Cs,DIrl, whete
z: )M(1)t. ùIoteover, Z f C5,11
itr M (.1)
+ I and a non-zero power of .9 js aû ;rlner automotphisû of T .
(2) (Prop.3.r l9]) Every monic intariant polyaomial f(1) € lll1,S,D)
can be written in the form J() : aa(z)U(t)*, where ù €T is invertible.
m ) 0 and u(.2)
is a monic polvnomial
in the cenier Cs,pfz)
ofTl1,S,Dl.
I
We will say that the center Z ol :Ilt, S, Dl is noo trivia.l :f Z + C
s,D.
In the foLlowing
lemma the notation will be as in Lhe above proposition.
Additionally we rvill assume that Z is non triviai.

Leûrrara 1.4, Let f (l) e Tl1
,
S, Dl be a monic non constant invarra.ni poly-
nomid and Z = Cs,Dlzl denote
the center of Tll,S,Dl. The followittg
cortdit ians ar e e
quiv
alent :
(i) l(t) car not be prcsented
is a p.odùci of two monic nor-cons,alli
i--".i"-i ^^À,n^-;"1.
(ii) Eitnerf(t):M(1) or tàere r's an int ertible
B e T suchthat
Bf(1) e
C5,pfz] is a monic irreducib-le
poll'nornid ir C5,6[,] diff"rent from
Proof. (i) + (;,). By Proposition 1.3, l(i) : ao(;)M(i)'" lor somc
invertible
a € ?, a monic polynomid ,,(z) e Cs,Dlzl and m ) 0. Since
both /(1) and ll4(l) are monic polynomials.
ao(z) is a nonic pol1,nomial
ln "lt,S,Dl a.nd, clearll, cvo(:) is rn inva-nrnr
pollnomial. Thrr"tore.
rhe assumpiion on l(i) yieLds
rhar either T(1) = M(1) or l(11 = ,ad(:)
Suppose
thar /(t) - ""(,), i.e. a-rl(t) : a(.2)
€ C5,p[.:]. Firsr rve
will shorv
th:rt a-rl(i) is irreducible as a poll'nomial
in C5.pl;1. -A.ssumc
tha.t
a-11(i) : l,,'(.')-:(,) for some ui(z) € C5,pl"l rvith des.'J.) > A,
i:1,2. 11".u;11 treat o;(z)'s as
polynornials
ir1ârlddenote ôr(l) = r.lr(z).
; = 1,2. B]'Proposition 1.3. the leading
coefÊcien[s a1,a2 of polynomials
t1(f), t,r(1)
are irvertible iû f and
a, r;r çi1.
a- r;13.1
ale Lnonic i .ani!ût
polynomials.
Therelore.,re côn
prcsent
/('l) in ihe forn
l(1) =,]r1(t)r'(1) = (a,1?
.ar
)(di' ùr
(i))(.t'., t ))
Since
/(t) is r]loûic.
ûa2a1 : I and the above eqtaliil' shows thai l(1)
can be decomposed into a pr-oduci of |lvo non-consiant monic irl.ariart
polynomials. This corcrariicts .Jur
;rssrrlpr.ion and esrablishes
.t rllll r5
an inciecorncosirble
poll'lornial in C:'
l. :1.
Rec:rll
th-r : = .\.lI(1)' i.r ..omc invertibie I e 7 and I -- 0. Lisiûg
ihis presentarion
of; an.i
ihc assumprion
on l(r) ii is eas,v to check thar if
plfl)-: for some irvetibie p 6 ? then
p : -l
anrL
I - f. i.e.
/({) = M(i).
Th.' "omt
i-r- h" troo of h- irro i -r;on ;) . qrrt.
(;;) .- (r). If f(l) - ,1{(J)
then ciearly
/(i) can not be decomposcd rnto
the product of lrvo non constant monic invariant pohnomi.rls. Suppose
'har rher" is cn,nr"rr,bl^ J C f.",.h -h"r B/(tt -so[rl . e morri
.
irreducible poll'nomial in z difcrent from z.
Lef 1 I lr(1), frlJ) a TF,S,D1 be monic invariant polynomials
such
that l(i) = fiit)l:ti) \li: will shorv thah lr(1) = 1. B1r maling ,rse
of
P:ooo irion 1..)
t. rn rvrirc polrnomial. /r l)./rr/\ rn h. iorm /,(/) =
'aiuiiz)M(.1)^ where.li € ? is invertible,
o;(z) is a monic
polynomiaL
in
C 5-plz),
m; ) 0: i : 1, 2. Then
BJ t) - Jdt't\z .\"1't - a;r, :.\/t t)tu2 = s,tz ':,z.aMIttût-'n'1

for some invertible
7 € ". Since
p/(f ), u1(z),
u2(z) are monic in z, central
polynomials, 1M(1)f,1+n' is a rronic central polynomial io .z. Using the
description ol central polynomials it is easy to see that IM(1)^'+^'z - ,k
for some
À à 0. Therefore
B1(1) : u{z)a2j)zr. Because
pl(l) is an
irreducible polynomial io C5,p[z], we get:
(*) ,[ = 0, since otherrvise
r,.,1(z)
: a2(z) = t anJ'
Bl(,1)
: z
(**) r..:1(:)
= 1 or u;r(z)
:1
Since
À = 0, i(r) - dir,ri(,),
i: 1,2. Now the conditiot (**) together
with the faci that /1(t) I 1 forces
r,.,2(z)
: 1. It means that fr(l) - 1 atd
establishes
the lemma. I
Up to the end of ihis sectioû we will additionally assume that 5 and
D commute. It is well knorvn ihai it this case 5 can be extended to an
automorphism of ?[t, 5, D] by séiting 5(1) : I aIrd D can be extended to
an S-derivation ollIlt,S,D) by D(l) : 0.
In lhe next lemma we will describe the set of all monic invariant poly-
nomials of minimal oon-zero degree and study the additive commutator
lM(i),11
Lemrna 1.5. Suppose
tÀat M(1),M'(1) € ?[1,5,D] are monic r.nvarianr
polynomials of minimal non zero degree. then :
(i) M'(i) = M(i)+c for some
c € f. Ïf clO then
c is interiiblc in
T and S" is an inner aut'oraorp,hism of T determined b,v
c. rvàere
î : .tes
M(j).
(ii) M(i) = t +h ii and only if D : D 6,5
(ie D(.r) = s('lô ô.r
iof
all
r eT).
If moreover 5(À{(i)) I M(l) tÀea -9
is an inner aufornorphism olT
and
"1t,5,
Dl
1 T[t']
(iii) M(r)i
=1M(.1)
jf either
des M(1)
> 1 or S(.M(t)):
M(t)
i!) lf M,t'l = lM t'rh"n rher.-;.'".€?suc-h rhe'5,c) =c. D r =
0 and
s(M(1)) = M(1) + c.D(M(1)) = cl.
Proof. (i) By Proposition
1.3 M'(i) : au(z) l"t
(t)^ for some
m à 0
where a € T is invertible and r.,(z) e C5,p[z]. Comparing degrees of
polynomials appearing in the abor-e equaliiy we obiaio that either m : 1
and M'(i) = M(l) or rn = 0 aod M'(i = du(,). In the second case- by
using again Proposition
1.3,
we gel I"tt(.1)=M(1)+clorsomec€7.
Now, for any r ç -R.
c1
: (,.M1@
- M({))r : s"(rXM',(i) -M(1)):s"(r)c,

whele lr : deg M(1). h mears that the eLement c norma.Iizes R. Now the
statemert (i) follows from the lact that nor-zero -R-normalizing elements
from T are iavertible.
Notice that in the proof of (i) we have oot used the assumption that .9
commutes with D.
(ii) Since M(t) : 1+ ô is inva.iatt we have, for a:ry o € T,M(1)Æ :
S(c)M(t) and a comparison of independa.ri terms on both sides of this
r_ ._4i.e.D=D ôs.
caua(rurr rcauJ ru u \,r ) T vt - u \t )t
Conversely
i D: D b.s
then we
easily
verify
ihat (1
+ù).r = S(.r)(1
+ô)
for any s € l and that (l +ô)l: (J
+4(l +ù) where ê = ù
- 5(ù). This
shows
rhat M(1) -1-b is invariant
Norv.
s(M(t))
- s(r+ô) - 1+s(à)
- l+ô+s(Ù) -ô = M(t)
-".
Hence S(M(t)) + M(.1) :f and only if c f 0. Since
"9(M(l)) is obviousll'
an inr-ariant polynomial of minimal non zero degree:
part (i) above shows
that if S(M(t)) l,jI,I(J) then 5 is an inner arromorphism of T induced
by
c ilnd one can check that c-1(1
f ù) is a central
polynomial
in ?11,5, Dl.
This yields
Tlt, S, D) - TVl fo|ii= c-r(i + ù).
(iii) &(iv). By (i), s(ttz(i)) = M(1)+c for some c € ?. Let a € ? be
such
ihat M(i)1 : (1
+ a)M(i) (M(l) is monic invariant). Then
M
(.1)1 = 1M(t +
a M(1)
=.e(M(i))r
+
D(M({))
+. M(1)
=
: (M(t)
+
c)1+ a M(1)
+ D(M(l))
Hence
(1) DlMt,tD -3 M(1), c1
lldesM(t) :' 1, ihensince
de9N(.Lf
(.1))
<.les LI(.1) thc cquation (1)sirols
that a = 0 i.e. V(t)t =tU(t).
1l des M(.1)
: 1 but s(M(i)) = M(l) then
c = 0 and (1) impLies
that c = 0.
This completes
part (iii).
Now il M(1){ : lrVI(i) the element a deflned a.bove
is equal to zero ard (1)
shows that D(M(1)) = -ci. Where
c € ? is such ihat S(M(l)): M(.1)+c.
If c I 0 parr (i) of this
lemma implies that S(c) : c and. by comparing
s(D(M(r))) and n(s(lr(t))), 1ve
set
,(c) = 0.
Example 1.6. Let ùs give an example of a monic invariant polynomial of
minimal degree
M(r) in an Ore exrension
?!,5, r] such that SoD: DoS
but l,{(i)i I {M(r). Le*ra 1.5 shorvs that the degree
of such a
polynomial
rnust be ore- Consider the polynomial ring Àlc] over a fleid  and let 6 =
c!./.4. Deflne an artomorphism over
R: Àlc]lù,6]
by putting 5
(2(.c)) = p(c)
for p(c) e Â[c] and 5(c) : b - c. It is easy to observe that 5 is a weLl
deflned automorphism of .R and ihat 5 o D-6,e : D-l;s o S. Let ? be ihe

symmetric
Martinda.le
ring of quotients (e.g 1: R if cha.r
È :0 since in
this case.&
is simple).
In the Ore
extension
?11,.9)D-ôis] the polytomial
{ + ô is inva^riant. Since
D-6 5(ù)
: -cù we get lô : (6
- c)l - cô
and so
(r
+
ù)r
- r' +lb
+
ct
+
cb
= (r
+c)(r +
t)
This shows thai (1
+ ô)t
l1(1 + ô). r
In the sequel
rve rvill use the following simple technical observation.
Lernma 1.7. Suppose
ihat' À € ? ajrd
g(r) e:fVtS'D) is not zero divisot.
Moth s(1)
and
l9(1) commut'e
with1, then 5(l) = f and
D(À) : 0.
Proof. Suppose
ihat 9(i) and l9(l) commute
wirh 1. Using
regularitl'of
9({) ii is stardard to see
thaL ) commutes
with 1. This implies the thesis.
The following iemma is of independent interest.
Lernma 1.8. Suppose
t,hai either
deg
M(1) > 1 or S(M(t)) = M(t) and
let 9(l) be a moûic in\,ariant
polynomial. Tlea 9(1) commutes witÀ i
Proof. If the
center
Z ofT11,
S, D) is trivial thet, by Proposirion 1 3, every
monic invariart polynomial is a porver of M(1). In this case the thesis
is a
consequence
of Lemma 1.5.
Suppose
Z is non trivial, iç. [ : Cs.ç'lz), rvhere : : ÀM({)l for sorne
:nr"r-rble
À f, 1
- 0. Bl mckirg us" of L.mma" 1.5
rnd i. rvo
g^-
s(.\)=ÀandD(.\)=0 tsl
Let 9(r) be a monic invariant
polynomial. Then. by Proposirion 1.3,
s(1) : de(z)M(l)^ for some lnvertible
a € T, u(;) € Cs,p{.21, rn à 0.
Since
M(l) is invariant.
Property
(*) shorvs that M(t).\: ÀM(1), hence if
rve rvriie
o(z) = !i-o a;:', a; € Cs,p we then get
ellI = o,,z,Mr,'- :^|",.1'-U,r,' -
À compariso;r cf leaCi;g coe{lcients shorvs that 1 = aa,,\' and so d is 5
and D irl,ariant. Hence
I commutes
with .r. u(z) and,thânks to Lemma I.5
(iii), also
with M(t). This ploves that I corrmutes
with 9(1).
r
Let us recail that an ideal I of R is called 5, D stable if 5(1) = f and
D(.r)
ç r.
7

Lemrna 1.9, Suppose
t.hat either
deg
M(l) > I ot S(M(i)) = M(t). Let
s(l) : Xr'-o cJr € ft,S'D] be eilJrer
a monic inrrariant
polynomial ot
g(t) e Z - the center of ?[t,5,D1. Lhen there is a non-zero
5,D sfable
id,eal
I of R suc.h
tfiat I9;, gil C I for 0 S ; S t.
Proof. Assume
that 0l / is an 5,D-stable ideal of E such
that 'I9(f) C
EF,s,Dl. Since
5(r): J, c(t)r = Je(i) c R[i's,D]. Now it is straight'
forward
lo verify
(cf. Lemma1.3
[9])
thatq;JCÊfor0!i !s There
fore in order to establish
the lernma
it is enough
to find a non-zero
5, D-
stable
ideal
I of R such that I9(l) c RF,S,D]. First we will frnd such
an ideal
for 9(1)
: M(l). By Lemma 1.5,
S(M(l)) = M(l) * c where
the element
c € T satisfres:
5(c): c' D(c) : 0 and c-R:.Rc. Define
f : {r 6 Rlr MO € R[1,5, D] and
rc € R]. Clearly
I is a left ideal of R
and I I 0 by definition ol î. Since
both ,ÙI(t) and c nonnalize
R, I is arr
idcal of E.
Let r € I. Since
5(c) : c" 5(')c : S(rc) € R and 5(r)n4(t) :
s(rM(l)) - s(r)c e R[t,5,
r]. This shorvs
that S(I) C I. Applf
ing ihe
samc argurnent
to
5-r rve
obtain
5(/) = f- By Lernrna
1.5
D(M(r)) = d.
Hence
D(r)M(l) = D(r M(l)) + S(r)cl € R[1,5'D], Since
D(c) = Q,
D(r)c e E. This completes
the
Proof
that
I is an
5,D-stable
ideal of Ê such that IM(r) C E[i,5,D] .(-)
If tire ccrrtcr
Z of Tlt,S,Dl is equal to Cs.o then, by Proposition i 3,
every rnonic invariant polynomial is a porver of M(l) Thrls tlle stalcmctrt
(*) yields the thcsis
in this câse.
Suppose
thal Z * Cs,p. Then. by ProPosition
1.3.
Z = C5,p[;l lh':rc
z : ^ùt(l\t for some
I > 0 and an inverlible element
I € î. By Lernûra
1.5,
M(l)l commutcs
$
ith l. Hence Lemma 1 ? shorvs
that 5(,\) = À alrcl
,(À) = 0. Norv using
the above
property of ,\ togetber
rvith the statcrnelrt
(*) it is easy
to cornplele
the proof for 9(i) e C5,p[;]
Finally let 9(l) be a monic invariant polynomial Theo, by ProposiLirxr
1.3, 9('l) :,'u(z)M(l)^ for some
m ) 0 rvhere a € I is invertibie
arrd
a(z) e Cs,olzl. Lemma 1.8
implies
that 9({) cornmutes
rvith 1. Since
the
polynornial o(z)Àf(i)' also commutes
rvith l Lemma 1.7
gives us 5(a) : a
and D(,r) : 0. UÀing this property and rvhat has been proved above f"'
M(l) and for central polynomials, it is easy to shorv that there is a non-zcro
5,D-stable ideai 1 of R such that I9(1) c Eli'5,Dj, as required.
I
Prirne ideals of R[t'S'D]. lu this part rve
give a description of prirnc
ideals of R[i,5,D] having
zero
intersection
rviih the coefficient ring l? I1s-
ing Lemma 1.1 it i3 standard to prove tbe follorving:

Propositiou 2.1. Fot tbe ring n[t,s,D] tlrc {o ovitg corrditir.rrs
a.r'c
eapivalent:
(i) 0 is tlre only R-disjoint ptime ideal of Rlt,S,Dl.
(ii) -R[],
S, D] has no non-zero R-disjoint idea.is.
(iii) 1'[], .9, D] d.oes rût coûtain non-cotrsfant monic in vatiant polynomi .
I
The equivalence given in the above proposition can be also expresscd in
terms of properLies
of 5 and D (cf. Th. 2.6 [9]).
Because of Proposition
2.1 rve rvill further assume thai È{1,5,D] )l. s
non-zero ll-disjoinb ideais. Notice that in this case there are ûôn-constan[
moric in!.ariâlt polynomials
in 1'[t,.9,D], so
M(t) I I.
5pec"(Ê[i,.9, Dl) will denote the sel of all prime ideals
of Ë[], S, D] rv[ic]r
alc E-disj oint.
Mas"(Rll,S,D)) rvill stand for the set of all maximal i<leals
arnong
It
disjoint ideals.
Since ,R is prime, it is easy to check that
Mar"(Rlt,s,Dl)
Ç
spec.(n[r, s, D]).
This observalion
rvill be used
freely
in the sequcl.
We rvill
conlinue
to use the notation from Propositiqn
1.3. In particrrlar
tlre center
Z o1'Ilt,S,Dl is non-trivial if it is nol contained
il T. In tlris
case Z : C
s,ol"l rvhere : : )M(l)' for some irvertible ) e î arrd 1
> -.
Tlreorerrr 2.2. For z non-zero ideal P ofTll ,
S, D] tÂe
follot irrg coldi tiors
iuc eqrir a.lel c.'
(i)
P €
5!ec"(rlr,s,Dl)
(ii) P €
n.f d'.("1i,
s, Dl)
(iii) P - f(l)"[1,5,D] vhere
f(t) € ?'li,5,Dl iseirJrereqrralroil(i)r,r
tlrc center Z ofTll,S,Dl Ânon trivial and tàere
is an invcriil'lc/ €
T sùch that
pJ() ç Z = C5.plt) is a rnonjc tncducible pt>lynontiiLl
(as a polynomial in z ) diferent frorn z.
Ptoof. (i) * (iii). Lel 0 + P e Spec"(Tlt, S, Dl). By Lemma 1.1,
rvhere
lp(1) denotes
We will shorv
that
P c lp(1)rl1,s,Dl ('l
lhe moric invarialt polyrromial
associated to P.
P : /p(1)
îlr, s, Dl. Define
= {Â(r)
€
T[r,s,
Dll/pr{JÀ(i]
€
PJ .

Since
fp(i) is invarianl, P. is an ideal of ?'li,S,Dl and, by Lcrnnra
1.1,
P"nT * 0. Clearly
we have (/p(l)T[], s, D])P" c lp(t)P. c P and
P" f P. Henc" primeless of P and (*) establish
lp(t)?[t, S,Dl: P.
The fact that non constanù mon;c invariant polynomials generatc trvo-sidcd
idea1s and primeness of P implies that lp(l) can not be decomposed inLo
the prodrct of two n<jn-constaot monic invariant polyuomials. Now Lemnlô
1.4 completes
the proof
of (i) - (iii).
(;t;) + (;t). Suppose that P = l(l)"ll,S,Dl, rvhere
/(1) is described as
in (iri). U Pl?) is a central
poll nomiai then the leadirrg
coeiicieût d of
p/(1) nornalizes
R, so
a is inverlible in " ancl
ir tpl(l) is a.
monic invarj:rrrl
polyrrorni:rl.
Using Lemrna 1-1, it is casy to sce
lhat a-\Bf(l) = lp\l)
Thus, by Lentma 1.4,
/p(1) can not be decornposed
irrto a product of t\',o
ror-conslanL moric irvariant polynomials.
Norv lel I be a oû-zcro T-disjoirrt ;dcal of :l'ii, S,
D] such tlral P a L
'lhen, by Lernrnas 1.2 and 1.1,
/p(l): À(i)//(i) for sornc rnorric
invrLi
aûh
polyrornial À(1)
€ "ll,-9,D]. This implies that lp(i) - //(1), si ,(
fr(l) I t and /p is indecomposable.
Using again Lernma 1.1 rve
havc
P c I c fr(l)lu,s,Dl: fp(t)1:lt,s,Dl: P. Thus P - I rurd P €
rf a3"("11, s, ,l).
(tt) - (l). This implication
isadircct consequence of prirrrcness
of I;ur,r
of the faci ihal for P e ,Uar"(f[t, S, n]) cverl ideal strictly contaiuiug lJ
has
a non-zero iûlerseciioû with T. I
Combinilg Theorern 2.? and Propositions
2.1 and 1.3
Ne get l[c foLloù
ing:
Corollary 2.3. Let Syec(Z\ dcrrolc the slt ,ri ul prir-rrc
t,lctl,: ol 't
lire ccnicr ,i'ff,S,D). llIere is ,lrIe ro 'rrc cr.'rrcspo.rrrir:rcc Lltrr,'rr
Spec"('flt,S,D)) an,|Spec(Z) e-rcepr| rire cr.sc
r;henl11,S,D) /rærorr .zcrr;
?-disjoinc idcals
arrd r\o
non-zcro po\rcr
of 5 is an inrrcr autonr,rr2lisrn
,rl
?. In r.lris case Spec(Z)
= {0} bui s?ec"lTi{,5,D]) = {{J,,Ir(i)1'[r,s,lJ]].
T
As : corr.cnrorre,rf Th..,r.m r.l and Lc:n:na 1.4 rve
also oLteirr tLr:
foLlowing:
Corollary 2.1. Let B; P ç,
Spec"(.Tlt, S.
Dl) rIeD P - lpl.t)'flt,S,D),
where I p(r) is i.he rnonic inr.zriant polynotnitl itssociiltcd to P . a
Now rve
rviil pzrss to the descriplion of 5pec"(n[i,5,Dl). [-or this sorne
preparation is nceded. For an ideal l of n{t,5,Dl we delinc the closule
{11
of 1 as /i (r)?[i,S.r]n R{i,s,Dl if I I 0; orhcrrvise
III = 0. \\'e rvill
sa1' that I is closed
if I : [Il. This nolion ças lirst introduced in [3]
for
l0

polyromia.l
rirgs. Using
Lemmas
1.1 and 1.2, it is straightforward to verily
that ihe followiog
holds:
Lemma 2.5. Let I, I be ideds of Rll,S,Dl. Then:
(i) r c [r]
(ii) Irr c J tÀen
III
c [J].
(iii) lll is cJosed.
(iv) If 1e Maî.(Rl1,S,Dl) f,hea
-I
is cJosed.
I
Notice also that every
P e 5pec.(7!,
S, D]) is a closed ideal of ffl,5, D].
since, by Theoren 2.2, every non-zero
prime ideal
belongs to M ar"(f V,
S, DD.
Lernrna 2.6. Sùppose
that every P € Spec"(,Rll,S,Dl)
js closed.
Then'
('i) If P+ 0 r-herr
P € srec. (RF, s, Dl) if ar1 d only if P e M ac.(Rlt, s, D)1.
(ii) TÀere
is one to one co.ffespondence
between Spec"(Rl1,S,D]) and
S p ec"(T
11,
S, D)'1
given
by
F : spec.(Rlt,s,Dl)
+ spec"(T
?,s,Dl)
and c : s p ec.(rfi,s, D))
- s
pec.(Rll,s,
D))
where for 0
f P e S
p
ec
"(Rl, S,
D]), F(P) : f
p(.t)ll
lt,s, Dl ancl for
P
e
Sp"c"\T
r.
s. Dl' G,F = F 'Rlr.S.
Dl.
Proof. (ii) First rve rvill shorv ihat the maps -P
atd G are rvelL-defined. Let
0+ P e spec"(R1l,5,D1) and l'(t), l:(r) € TF,s,Dl be monjc invariant
poiynornials such that lp(l) = fi(t)l:(i). De{ine I; : /iil)"11,5.D] .
Êt1,.9,r]. i:1,2. Then clearly
P = [P] C Ii, r = 1,2. \Ioreover-. using
closeoess
of P. one can show lhat IrIz C P. Ilencc. b1'primcness.{ P.
either 11 : P <t 12
= P. It mea"ns ihat either lr(t) or lr(t) is equal to
lp(t). This shows thac lp(l) can not be decomposed into a product of
trvo non constrant
monic invariant polynômials. Norv Theorem 2.2 together
wirh Lcmma 1.a
vi"ld F(P) IpttYrl,S.D) € Spec.allt.S.Dl\. i... F is
well-defined.
Now take 0 I F e Spec"(.lflt,
S, Dl). By Theorem
3.J a:rd Lemma L.1,
P: fF0)1:U,S,D) a:rd
lp(l) can
noi be
preserted
as a
produci
of two ron-
constant moric invariant polynomials. Let 1 € Mao.(-Rlr,5,D]) be sLrch
that G(F) C I. Then, by Lemma 1.2,
there is a monic invariant polynomiai
à(1) € rlr,S,Dl such that l6ipy(i) = h(.1)fr(.1).
Since
161p)(l) : /p(i),
Àlr)=lrîd lclp't = /i rr. )'lorvG'Fr - I follorvs. L"caus. borh
Ûr P
and f are closed
ideals. This shows
that
GtP)
€
Mar.(nF,.ç, Dl)
11
(')

Theo clearly
C(F) e Srec"(R[i'S, D]), i.c. G is rvell delined.
I(norving that F and G are rvell-deû-ned it is siandard lo complcte ilre
proof of the statement
(ii).
(i) The inclusion
Moc.(n[t,S,Oi) C Spec"(R[i,5,D]) is clear.
Ler0l P e Spec.(Elt,5,
Dl). Thcn, by
(ii), P : G(P) for some suilable
F e Spec.(ttlt,S,Dl) *a (-) yields
P e Moc.(R[t,5,D]) r
The al>ovc
lemrna
Logether
with Theorem 2.2 provide a dcscription of
S!ec.(,q[{,
5,
D]) in the case
wheIr evert P e Spec"(-R[i,5, D]) is closecl.
Notice tlrat every P € SPec.(,f
lt,S,D)) is a closed
ideal of Tll's,Dl
since,
by Theorem 2.2, every non-zero prirne ideal of Tll,5,D] belongs
to Mcc.(?[i,S,D]) and by Lemma 2.5 (iv) every
P € Mo'"(?'[1,s,Dl)
is closcd. In particular if I? is a symmetricalll closecl
prime riILg (i e. if
R: ?) thcn ihe condilions
of Lernm:r'
2 6 are
satislied We rvill lorv sir'"r
tlrat evert P e Spec"(Rll,S,Dj) is closed
if one of the follorvitrg
conditiorLs
is ful-frlled :
1) R is lefi and right ncciherian
2) .R satisfres
the dcscending
chain conclition on irvo sided ideals
3) 5 and D commule
arrd
either
deg M(l) > I or S(ÀI(i)) = Àf(t)
Proposition 2.?. Suppose
-R
rs
left :urd rigùt Dr:tLerian TJrcn
ever.r' r,J.e?
P e sp"c"(R\t,s. D
i) is clo"ed
Proof. In virlue of Lcmma 2.5 (iv), il is crrough to shorv tlrli
spec.
(.1?[t; sD])
ç À'rd'"(nlr
'
s' rl).
Let s be llLc set of regular
elen.ItLs
in It and Q - 1?5-t = s-'lt l'r'
ihe classic:rl
lcfi and right .trroti(rnr
Iiu,l Sint:c
l3 is plinrc. Q is;r lifr
rncl riqlLl ;rr',iuian sirnt,le lilre bl gol.lic: tlr.1)rcrn. l'. irj:iirnr(Lir:(L
ri) "r':
lcnd borh 5 :rurl D to Ç ancL
to prole thar S is bot.ir a riglrt and lcir
clenominaLor
sei in,4[1,5,D] such rhai S-t(n{t,5,D]) = nll,5,r15 | -
Ç[i,5,D] (cf. [6] Lerrunas
1.3
atd 1.{). Sincc
n[t,s,l] is borL lcfb;Lrrrl
right no:therian there is a (1,1) colresp<xrrierrce
betrvecn
tLe sets {P e
spe4.R[1, 5,
r])lP n.J = O] and
spec(Q{i, s, Dl) (cf.[lt] Proposition
2 1 I {i
(vii)). Sincc Q is simple and hence irlso svtrmclricirily closed
rve h;rve
Spec(Ql1,S,Dl = Spec"(QItis'Dl) = rl'ro'.(Çti'5,Di), rvhere the last
equaiiLy cornes
from bhcorem
2.2.
On ihc orher hand if Pe Spec.(n{t,S,l]) thelr
ohviousl.'-
PnS:à
aûd lhc iuclùsion
.9pec.(R[t,5'D]) Ç ,1Ioc.(It[t,5,D]) is uorv arxl c:r-s1'
consequencc
of lhe fact thah the (1.1) correspondence
rnenLioneci
above
preserves inclusion.I
Proposition 2.3. Suppose
tfrat E -"alisfies
tl.c.c. ott trvo-sirled
rdcals
T)rcn
every idcal P e Spec"(R(i,5,
D]) is closcd.
l2

Proof. Let 0 + P e Spec"(Rll,S, D]). First
we rvill
Ând
a. non-zcro idcal
I of R such
that I[P] C P. For doing this, let us define
Q : {À(t) e
?[l,s,D]
l/p(/)À(1)
€ Àlr,s,Dl] and
r : {r € Ëltp(l)r € n[t,s,D]].
Clearly
f is a non-zero ideal
of R and [Pl : lp(l)f[l,s,D] o R[1,5,D] =
tp(f)Ç. lvith the above totation 1ve rvill prove that:
for an5
m ) 0 and A(l) = ),a,t' e Q
r=0
.r-,1('
)
c Ê[r,s,D]
(-)
Let
À(r)
= D- o
otti
e Q. Since
/p(l) is monic and
fp(r)À(l) € Rli,5, Dl,
a* € Ê. This eslabiishes
(*) for rn
- g.
Assurrre
rn > 0. By above, a- € It. Thus
J
P(t)Ia^t^ c fP(l)Jt"'c Rll,5,Dl
Using
this inclusion it is easy to see tirat
J(À(l
)- o-l- ) C Q, Therefore,
b1'
ioductive
hypothesis,
f--rJ(À(t) - o,"i^) c E[i,5,D] and the statemeùt
(*) follorvs.
E is a prime
ring
lv;th
d.c.c. on trvo-srdcd
i(l"als, lhus J'n - Jn+1 --
J f 0 for so,ne
- > 0. Therefore,
Lr) (-), tÇ c R[l,s,.D]. Defirre
1 = {r É
5"(t)l'lp(l) € P, n : deg
f pJ)\. TheD. b1'
pri:neness
of R, the icleal I is
ron*zero.
Since
//p(l) c P arrd.t-"(I) q J rvrth n : deglp1i.1, rve
lre,;e:
r2ln1
- t')1,'ç1q
c (r/p(r))(s-"(r)Q)
c Pnlr, s, Dl
c P
Let A:.Rlt,5,Dlr'?À{l,s,Dl. Then,4 is an idcal of l?lt,5,Dl Lnv,nLs
non zero inLr:rsecLion
rvith R and. b.'- thc above,
A[P] C P. Norv
plinrc:r"',s
of P implics [P] = P, i.e. P is closcd. This esiablishes
tltc propositior. I
Now rve
rvill invesLigaie the case
rvhen
5 ônd D commuLe. For this rvr:
rvill use
a subring
T" of T corrsisting
of all such elernents
9 € ? tlttrt thcre is
a non-zero 5, D-stable ideal
I o{ -R
such llat Iq, qI C R (one can look at
T" as a lvlartindale symmelric qùotient ring of .R constructed rvith respecr
to the fllter of all nor-zero 5, D-stable ideals of 'R ). It is ea-sy
Lo scc thrt
s(?1") : "" and l(".) C 1.. Therefore rle ca. cotsider the lollorvilg Orc
extensions:
.R[i, 5, D] c ""[], S, D
I C al
11,S'
D).
We rvill continue to denote by ,LI(i ) € "ll, S, Dl a monic invariâtt polyno-
mia.l of ninimal non-zero degree. Às rve
remarked earlier, such a. polynomiai
exists
of Spec.(È[], s, D]) I {0}.
13

Proposition 2.9. Suppose
that S and D commute
and that ejther
des M(1)
> 1 or
.9(M(t))
= M(1) Then every
ide P e sPec"(Rlt,s,Dl)
is closed.
Proof. Let O + P € s2ec"(nft,
S, D]). ,9 comnutes
with D, thus we
can apply Lemmas
1.8 and 1.9 to the polynomial fP(r) getting lP(i) €
I:"ll,S,Dl and lp(i)r = tlp(t). Now, using the fact ihat ËFJs,Dl c
f.l1,S,Dl and lp(1) € ?'.|i,5,D], one can easilv
check that both
f
p(l).l"l1, S, D) and
lp(1)f lt,
5,
D] have the
same intersection
rvith R!,5, Dl
Therefore
in order to prove ihat P is closed.
il is enough
to show that
P : I p(1)1:"11, S, Dln -RF,
S, Dl. We
will do ihis in two steps.
First we will
establish
the following:
P
- lP(l)ÊF,
s,
Dl
n Rll,s,DlcP (.)
Consicler
P. : {À(1)
€ ÊF, S, Dl I
lp(l)àt) € P}. CLearly
both P ancl
P. are non zero righr ideals of R[i, S, D]. Since
lp(l) commutes rvith I anrl
lp(1) norma.Lizes
i?. /p(1) also
normalizes
-R!,5, D]. This implies
thar P
and P" are
ideals of R!, '9'
D] Notice ihat PP" c P but P" is not cottaired
in P, becaus"
P.^ F 70. Norv
prim"ness of P yi"ids che
staremenc f
'\
Let
e(1)
€ /p(t)r.lt,s,ll n nli,.e,Dl. Then
e(l)
: /p(l)È(1)
for some
À(l) € ?.F,5,D] and.
by defrnition
of ?", there
is a non-zero
5,D-stabLe
idea.l
J of -R
such that rh]) c Rll'S,Dl. Since
5(/) = J and
lp({) is
invanant,
we have
J/p(l) = lp(l).I. There{ore
Je(i) - /lP(l)À(r) : JP(.1)Jh(.1)
c
c lP(r)Ê[r,
s, D] n
nll, s, Dl
=
P
5,D stabilitl'of J l ieicis
also ;111e
,r
: "I
R[1, S' D] = n[i,5, D]"r is an
icleaL
of R[l,5,D]. Using
this logethe!
lvith (**) alrd
l') we
ijel
t(-R[i,
s, D]e(r)Êl1,
s,D))
c Rl1,
s.
D)I
s(l)EF,
s, Dl
c
c Rl1
,
s
,
DIP Rl1,
s, Dl
c P
cP
(-. )
Because
in R I A,
J is not included
in
e(t)
e P. Thus
fu(t)r.lt,
s, Dl . Rl1)S'D)
the proposition.
I
The last three propositions together with Lernma 2.6 and the
preceding Proposition 2.7 give us imme<liately the following:
P and primeness of P implies
C P. This gives the proof of
rernarks
Tlreorem 2.10. Suppose
that one of the following condiiions is satisrSed
a) R is symmetricaily dosed (i e. T = R)
b) Ë is ncetÀerian
c) R sat.isÉes
D.C.C. on two-sided
ideals
d) S and
D commuie
and eit-her
M(l) > 1 or S(.M
(.t)) = M(1)'
14

Then every P ç Spec"(Rll,S,D]) is cjosed aid fot a non zero R-ùsjoint
ideal P of Rl1,S,Dl the following conditions are equivalent
:
(i) P €
sP€c"(-aF, s, Dl).
(ii) P
e
Moa"(Rlt,s,Dl).
(iii) P: f
(t)rF,S,Dl.nF)S,Dl
where
the
polynom)ai
/(l) €
?lt,s,Dl
is as descrjbed ia Tbeorem 2.2 (iii). I
Let us make a lcrv
Ênal
colomenLs :
1) If D :0l,'e can choose
M(i) : { so that S(M(i)) : M(i) and ihe above
theorem (case d)) applies.
2) Simila.rly,
if 5 = id rve
obviously have S(M(1)) : M(l) and condition d)
of the above theorem is satisffed.
3) NIor" g-nerally if 5 : r\e inner automorphism l. of ,R
indrr""d b:, ar-
invertible eleorent c e R or if D = Da:s for some ù in -R then standard
changes of
variables
sholv that RF, 5 : I.,Dl = -R[r',
Di
] and
]?{1,
.9, D6
5] :
-R[i'l,
Di'] and hence lhe above theorem stilL
app]ies.
4) We expect ihe conclusions
of theorem 2.10 above to be true rvhen J and
D commute bui ore case is missed
: the case
wher SD : DS, deg M(l) =
1,
S(M(i)) I M(t) and neither.9 nor D is inner or R, we cannot flnd ar
example satisfying all these
conditions.(Noiice ihat Lemma 1.5
(ii) shorvs
thât ir such ar exâmple both 5 and D are inner on T).
5) The resulls of ihis secbjon
suggest that the same decription of
<--^t tDû < nr\ -- i- rL^ -L^.,^ rheorem
shoutd hold for arbitrary Ore
extetsion
Ê1l,.9, D]. \oticc that such description exists iî and
ouly if every
P ç Spec"(Rlt,S,Dl) is closed.
-\cKnowleogment
The authors are verl'grateful to ihe referee for his helpful commenis td
suggestiors.
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