I nternat. J. Math. Math. Si.
Vol. 4 No.
(1981) 109-117
109
SOME CONDITIONS ON FIXED RINGS
EDNA E. REITER
Department of Mathematics
University of Cincinnati
Cincinnati, Ohio 45221
U.S.A.
(Received February 12, 1979)
ABSTRACT.
This paper deals with a question about the ascending and descending
chain conditions on two-sided ideals.
Using the ideas of the skew group ring,
certain results for two-sided ideals are proved.
KEY WORDS AND PHRASES. Fixed rings, acending and dcending chain condon
or two3sided ideals, and skew group ring.
198o
i.
MATHEMATICS SUBJECT CLASSIFICATION CODES:
INTRODUCTION.
Let R be an associative ring with an identity element i, and let G be a group
of automorphisms acting on R such that
IGI,
In this
the order of G, is finite.
G
paper we will prove several relationships between R and the fixed ring R
{r e R
r
g
r for all g e G}.
We answer the question raised by Fisher and
Osterburg [7], that is, we prove that if R has no
IGlr
0 implies r
0), and
IGI
torsion (for r e G,
if Soc R denotes the left (or right) socle of R,
Ii0
E.E. REITER
then we have (Soc R) [
qoc R
Thus for a semiprime ring with no
IGI-
G.
G
R
torsion, Soc R If
G.
RGc
Soc R
We also answer a question about the ascending and descending chain
conditions on two-sided ideals.
IGI
IGI-I
It is well known that if
R (i.e., if
R) and if R satisfies the ascending or descending chain
is invertible in
condition on left (right) ideals, then R
G
also satisfies the same condition.
However, the techniques used for one-sided ideals have not been applied to twoIn this paper, we prove the analogous result for two-sided ideals
sided ideals.
by using the skew group ring of R and G,’R,G
@
Y
Addition in R*G is
Ru
h)
componentwise- multiplication is given by the relations (ru)(su
g
)Ugoh
The inverse question, posed as follows, remains open: If R is
extended linearly.
semiprime and
(rs g
IGI -I
G
R, and if R satisfies the ascending (descending) chain
condition on two-sided ideals, then must R satisfy the ascending (descending)
chain condition on two-sided ideals?
The
question is true for one-
analsous
sided ideals.
For a left and right Noetherian ring, the Artin radical has been defined
If R is semiprime and
as the sum of all Artinian left ideals of the ring.
R, then A(R)
is defined if and only if
A(RG) is defined.
of this paper, we prove that under these conditions,
2.
In the final section
G
A(R) R
G.
THE SOCLES OF R AND R
First, consider the following theorem relating R and
proof is a variation of the proof of Maschke’s theorem.
RGymodules.
The
Note that all R,G-
modules are certainly R-modules; on the other hand, the left R*G-submodules of
R are the G-invariant left ideals of R (I
/
that I
g
c
I
J
g
G).
Let R be a ring and G a finite group of automorphisms of R,
THEOREM 2.1
and let @ N
R such
M
/
N’
/
0 be a short exact sequence of left (right) R’G-
modules such that the map m
/
IGlm
is a bijection on M.
Then if the sequence
CONDITIONS ON FIXED RINGS
iii
splits over R, it also splits over R,G.
PROOF.
See Fisher-Osterburg [7, Theorem 1.3].
IGI -I
They assume
R, but
the proof can be carried out easily without this as long as for any m
m’
there exist
n’
and
IGlm’
in M and N respectively such that
m and
M,
N,
IGln’
n.
From this, we obtain:
COROLLARY 2.2
M.
Let M be a left (or right) R’G-module with
IGI
a blJection on
If M is semisimple as a left (or right) R-module, then M is R,G-semisimple.
PROOF.
Immediate.
Now we can apply this to solve the socle question.
THEOREM 2.3
Let R be a ring with no
left (or right) socle of R.
IGl-torsion,
and let Soc R denote the
G
R c Soc R
G.
Then Soc R
We give the proof for the left socle only.
PROOF.
Note that Soc R is a G-
inarlant left ideal of R; thus it is a left R,G-module.
= m.
s
ft
biJection on Soc R.
Th =no b =h=
of R.
We can apply Corollary 2.2:
.
The simple components
y
GI
GI
o
a
Soc R is R,G-semisimple, i.e.,
n
=i--
Soc R
@K
i’
where each K
We now claim that each K
c
that X is a left ideal of R
G) ___c Ki;
R
RG
RX
.
i=
I’1
G.
G
R is a minimal left ideal of R
X_c
such that 0
this implies RX
Ki.
RG"
K.I
Then 0 #
GIRX
However, the fact that
For, suppose
RX__C R(Ki
RX forces
X, and the claim is proved.
+/-n
xi"
(i!%[ @ Ki)
f.+/-hd if
Thus,
direct and each
REMARK.
n
RG
Now, (soc R)
an
i
is a minimal G-invariant left ideal of R.
i
i
xig-
x
.
i
/
RG"
=a. show ha
j__ xj g- xj
is G-invariant.
Each K
i
n
I
( (R) Kl
C
e
extra assumption that
GI
n
(K C
O, i= l,...n, since the sum is
These equations show that each x
i
This theorem was subsequently proved by R. Diop
-I
..
G
G
R is a minimal of R
R, by M. Lorenz [9].
E RG.
[6], and, with the
112
E.E. REITER
We note the following corollary.
Let R be a semiprime ring with no
COROLLARY 2.4
G=
Then (Soc R)
G.
Soc R
By a theorem of Fisher and Osterburg [7], if R is semlprlme with no
PROOF.
G
then Soc R
Gl-torsion,
3.
Gl-torsion.
c__
Soc R.
CHAIN CONDITIONS ON TWO-SIDED IDEALS.
To deal with two-slded ideals in R and R
G,
we again consider the skew group
In particular, we will use the following lemma.
ring.
LEMMA 3.1
Let R be any finite group.
If R has the ascending (descending)
chain condition on two-slded ideals, then R,G also satlfles the ascending
(descending) chain condition on two-slded ideals.
PROOF.
G
{gl
rlUg i +
I I,
The proof follows the method of the Hilbert Basis Theorem.
+
12
lj
gn },
I
r
u
{r R:
R’G, we define Ik
+
k i gk-i
let
The n chains of ideals I
ru
gk
I} for k
th
i
for some
be the k
I,
12,,... lj,,...,
R,
For a chain of ideals
,n.
associated ideal of
lj,k
rl,...rk_ I
For
i,
I. as defined above.
3
n must each terminate;
this forces the chain I. in R,G to terminate.
It may be mentioned that the above lemma is true for any ring T and overring S such that there exists a finite subset of S,
n
S
Ts
i=l
i
and s.T
{Sl,
Sn }
such that
Ts..
z
It is well known (and easily proved) that for any ring S that has the
ascendiug (descending) chain condition on two-sided ideals, any subring of the
form eSe where e
2
e must also satisfy the same condition.
We use this fact,
together with Lemma 3.1, to prove the following theorem.
THEOREM 3.2
If
IGI -I
R and R satlfies the ascending (descending) chain
G
conditions on two-sided ideals, then R
satisfies the ascending (descending)
IGI -I
R and R has a composition
G
series of two-sided ideals, then R has such a composition series also.
chain
condition on two-sided ideals.
If
113
CONDITIONS ON FIXED RINGS
PROOF.
that e
2
e
isomorphism r
e.
(I/IGI) 7 u.
Consider the element e
G
e(R,G)e.
Also, we claim that R
/
(rUgl)e
where
6 R,G.
It is easy to show
G
To prove this, use the ring
G
gl
is the identity of G to obtain R
RGe
e(R*G)
Then, since R,G has the ascending (descending) chain condition on two-sided
G.
ideals, so does R
In the preceeding Theorem 3.2, we may weaken the assumption that R has no
GI -I
R to the assumption that R has no
Gl-torsion
for the cases of the
descending chain condition or the composition series property.
since
IGI
is in the center of R and
RIGI k,
k
is a descending chain
1,2,
of two-sided ideals which must terminate, forcing
This is true
IGI
to be invertible in R.
We state this as the following corollary.
COROLLARY 3.3.
If a ring R has no
IGl-torsion,
then if R satisfies the
G
descending chain condition (has a composition series of two-sided ideals), R
must also satisfy the descending chain condition (have a composition series of
two-sided ideals).
PROOF.
Evident from Theorem 3.2 and the paragraph above.
We conclude this section by noting an example due to Chuang and Lee [3].
This is a commutative Noetherian ring R with involution, such that R has no
2-torsion and the subring generated by the symmetric elements is not Noetherian.
In a commutative ring, an involution is an automorphism of order 2, and the
subring generated by the symmetric elements is the fixed ring.
This example,
then, shows that for the ascending chain condition, the hypotheses of Theorem
3o 2 cannot be weakened to assume merely no
4.
THE ARTIN RADICALS OF R AND R
G.
In a left and right Noetherian ring R, the Artin radical A(R) is defined
as the sum of all right ideals of R that are Artinian as right R-modules.
This
radical turns out to be a two-sided ideal of R; it is also equal to the sum
of the left ideals of R that are left R-Artinian.
i14
E.E. REITER
It is easy to prove that if R is left and right Noetherlan and if
G
then R is left and right Noetherlan.
then R is left (right) Noetherlan.
G)
defined if and only if A(R
and if R
G
is left (right) Noetherlan,
Thus, for R semlprlme with
IGI -I
R, A(R) is
In this section we prove that under
is defined.
G
these conditions, A(R)i R
R,
Also, Farkas and Snlder [5] have proved
Gl-torslon
that if R is semlprlme with no
’IGI -I
A(RG).
To prove the first inclusion, we need several preliminary facts.
First,
note that if I is any G-invarlant ideal of R, then G (or a homorphlc image of
G) acts on R/I by g(r + I)
g(r) + I.
IGI -I R,
(R/I)G RG/(I RG).
it can be
Furthermore, if
shown (see Fisher and Osterburg [7]) that
Also, we require an important theorem due to Bergman and Isaacs [i], which
we state below.
The trace of an element is defined by tr x
[
x
g’,
the trace
gG
of an ideal L is {tr x: x
(Bergman-Isaacs)
THEOREM 4.1
without
(i)
(ii)
L
L}.
G -torsion.
Let G be a finite group acting on a ring R
Then:
G
If R is semlprime, then R
is semlprlme.
0, then L
0.
Now we can prove:
THEOREM 4.2
PROOF.
Let x E
GI
If R is semlprlme and
A(RG).
rRG [A(R) I RG];
Only one inclusion requires proof.
that is, x
applicable so we have A(R)x
0, i.e. x
To prove the theorem, we show that
G
R and [A(R)
(A(R)x)
is a G-invariant left ideal of R and tr
G.
G
-I 6
R, then A(R) ] R
First, we prove the following fact about right annihilators:
[rR(A(R))]G rRG[A(R) RG].
of R
_
If R is semiprime and L is a G-invariant left ideal of R such that tr
0.
i
RG]x
O.
Now A(R)x
The Bergman-Isaacs is
[rR(A(R))]G.
A(R) RG
is an Artlnlan right ideal
By a theorem of Lenagan [8] if I is an ideal of a left and right
115
CONDITIONS ON FIXED RINGS
Noetherian ring such that I is left R-Artinian, then
Thus,
R/rR(A(R))
Since
is an Artinian ring.
IGI -I
R/rR(1)
is an Artinian ring.
R, it follows easily that
Using this together with the
the fixed ring of this factor ring is Artinian.
facts on the preceedlng page, we have the Artlnlan rings:
Ir.R )]
G
G
R
RG/(rRG[A(R)/
And, from the first part of the proof, we have
G]
R
is Artlnlan.
It is true that a right ideal I of a ring S is contained in A(S) if S/r(1) is
Artlnlan (see Chatters, Hajarnavls, and Norton [2], the proof of Theorem 1.3).
Applying this, we have A(R)
G
R c
A(RG).
To prove the other inclusion, we need the following theorem.
Its proof is
variation of a proof by Cohen and Montgomery [4].
Let R be a semiprime ring, left and right Noetherian with
-IEORI 4.3
R.
G)
A(R
If A
G
IG1-1
then RA is an Artinian left
is the Artinian radical of R
R-module.
Since R is left and right Noetherlan, there are left ideals of R,
PROOF.
K
maximal in RA.
And, since
We prove first that K
G
G)
Soc (R
A is artlnlan, we may choose a finite family
A
kn, to minimize K
of these, kI,
socle of R
(RG)
0.
(K
I/
Kn )"
For, suppose not.
is essential as a left ideal in A.
Since
G
Also, sinc
A is Artlnlan, the
is semiprime,
is a semlslmple Artlnlan ring and has an identity element i
S
Now,
ifK0:
G
Soc R
K
Consider R(I
S
e)
RA.
(Soc
RG)e,
2
where e
e
# 0
R(I S
The inclusion is proper since the sum Re
c RA is direct, and equality would force Re
we can choose a left ideal of R,
K0,
0 and hence K
so that K
R(I S -e)__c
0
K0
0.
Therefore,
is maximal in RA and:
RA
e)
116
E.E. REITER
We now claim that el
K
n)
0.
__
To prove the claim, suppose that e
Note that the sum A
K0.
S
G
R
G
A1
Thus, RA
S.
A1
A (i
1 S, so that Soc R
obtain A
R1
i
0.
n
Now consider
i=l g G
ideal of R and L
A c K
L
G
R
Since L
0.
A(I
S
Kig
L
RA.
G
R c L
0"
Kn
I
and
Then obviously
is direct, and that Soc R
G
Since Soc R
K0,
is essential in
A, we
a contradiction to the choice
Certainly L.is a G-Invarlant left
I
RG
G
A f’, R
G)
0, certainly tr (L
K
K
(K0
G
S
and RA
Also, L
0.
i
0.
S
K0,
s
of K
L
. ...
properly smaller that K, contradicting the choice of KI,
thus forcing K
1
G
which will make the ideal R
K0,
R
0.
c RA
G
R
A.
We now have:
0
By the Bergman-Isaacs theorem,
Thus, there is a natural R-module embedding of RA in the finite direct
sum of simple left R-module,
I
+/---
[ RA/(Kig).
Now it is immediate that RA
gG
has finite length as a left R-module.
We close with two corollaries.
COROLLARY 4.4
R.
G)
Then A(R
PROOF.
Let R be left and right Noetherian, semiprime with
c A(R).
COROLLARY 4.5
G)
PROOF.
G) is left Artlnian;
A(R).
A(RG) c RA(RG)
We have proved that RA(R
all Artlnian left ideals.
Then A(R
A(R)
GI -I
Thus,
A(R) is the sum of
let R be left and right Noetherian, smiprime with
G.
R
Theorem 4.2 and Corollary 4.5.
IGI --I,
R.
CONDITIONS ON FIXED RINGS
117
REFERENCES
i.
G.M. Bergman and I.M. Isaacs, Rings with fixed-point free group actions,
Proc. London Math. Soc. 27 (1973), pp. 69-87.
2.
A.W. Chatters, C.R. Hajarnavis, and N.C. Norton, The Artinian Radical of
Noetherian Ring, J. Austral Math. Soc. 23 (Series A), (1977), pp. 379-384.
3.
C.L. Chuang and P.H. Lee, Noetherian Rings with Involution, Chinese J. Math.,
no. I, pp. 15-19.
5__(1977),
4.
Miriam Cohen and Susan
Canad. Math.
5.
Daniel R. Farkas and Robert L. Snider, Noetherian Fixed Rings, Pacific J.
Math.
6.
Montgomery, Semi-simple Artinian Rings of Fixed Points,
pp. 189-190.
Bull. 18(2) (1975),
69 (1977), pp. 347-353.
Raoul Diop, Actions de Groupes, Doctoral Thesis, L’Universite de Poitiers,
1978.
7.
Joe W. Fisher and James Osterburg, Some Results on Rings with Finite Group
Actions, Ring Theory: proceedin_s of the Ohio Univ. Conf., Mrc] Dkr,
(1977), pp. 95-111.
8.
T.H. Lenagan, Artinian Ideals in Noetherian Rings, Proc. Amer. Math. Soc.,
51 (1975), pp. 499-500.
9.
Martin Lorenz, Primitive Ideals in Crossed Products and Rings with Finite
Group Actions, Math. Z., to appear.