Download STRONGLY PRIME GROUP RINGS 1. Introduction ([8], Conjecture

NEW ZEALAND JOURNAL OF MATHEMATICS
Volume 22 (1993), 93-98
STRONGLY PRIME GROUP RINGS
R .K .
Sharm a and
J.B.
S r iv a s t a v a
(Received M ay 1992)
Abstract. It is proved that if R is any strongly prime ring and G is any one-relator
group then the group ring R G is strongly prime, if G is FC-solvable then R G is
strongly prime if and only if R is strongly prime and G contains no nontrivial locally
finite normal subgroups; if G is a group having a finite normal series with F C hypercentral factors then R G is strongly prime if and only if R is strongly prime and
the locally finite radical of G is trivial; if K is a field then G can be imbedded into
an infinite simple group G* such that K G * is strongly prime. Some examples are
also given.
1. Introduction
A ring R is said to be right strongly prime (henceforth called strongly prime) if
every nonzero two sided ideal of R contains a finite subset whose right annihilator
is zero. These rings were first systematically studied by Handelman and Lawrence
[8], They raised the problem of characterizing strongly prime group rings ([8],
Question 3, p. 222) and conjectured the following:
Conjecture. ([8], Pages 215 and 222) The group ring R G is strongly prime if
and only if R is strongly prime and G contains no nontrivial locally finite normal
subgroups.
The main aim of this paper is to settle in affirmative this conjecture for the class
of one relator groups, FC-solvable groups which include solvable by finite groups
as a particular case and the groups having finite normal series with factors F C hypercentral. When R — K is a field, we determine some more groups G including
certain wreath products for which the corresponding group algebra K G is strongly
prime.
Domains (not necessarily commutative) and prime Goldie rings are strongly
prime. It is proved ([8 ], IILl.(a)) that if the group ring R G is strongly prime
then R is strongly prime and G contains no nontrivial locally finite normal sub­
groups. Hence proving the conjecture in affirmative simply means proving the
converse of this theorem. Strongly prime rings are prime. Connell [3, Theorem 8]
has shown that R G is prime if and only if R is prime and G contains no nontriv­
ial finite normal subgroups. For a study of strongly prime ideals, we refer to the
beautiful paper by Ferrero et al. [5].
2. Preliminary Lemmas
Throughout this paper a ring will mean an associative ring with identity. For
a normal subgroup H of a group G and for an ideal I of a ring R, we shall write
H < G and I < R, respectively. X is a normal subgroup of Y or an ideal of Y
in the notation X < Y will be clear from the context. For a group G , let L(G)
1991 A M S Mathematics Subject Classification-. 16N99, 16S34
94
R.K. SHARMA and J.B. SRIVASTAVA
denote the locally finite radical of G, that is, L (G ) is the unique maximal locally
finite normal subgroup of G. Clearly L (G ) is a characteristic subgroup of G and is
the union of all locally finite normal subgroups of G.
The following two lemmas are straightforward.
Lemma 2.1. Let R be a ring, G a group and H < G such that I fl R H ^ 0 for
every 0 ^ I < R G . Then R G is strongly prime if R H is strongly prime.
Lemma 2.2. Let H be a subnormal subgroup of a group G. Then L{G) = (1)
implies L (H ) = (1).
A group G is said to be F C if every element of G has only finitely many con­
jugates in G. In a periodic FC-group every finite subset is contained in a finite
normal subgroup.
The following lemma plays a crucial role in our further work.
Lemma 2.3. Let R be a ring, G a group, H < G such that L(G ) = (1) and G /H
is a periodic FC-group. Then R G is strongly prime if R H is strongly prime.
Proof. If W < G, then R W is prime by [3, Theorem 8]. This is because L (W ) =
(1) by Lemma 2.2 and R is prime, since R H is strongly prime.
Let 0 7£ I < R G and 0 ^ a =
ri9i e
Let N / H be the smallest normal
subgroup of G /H containing the cosets {g iH , g2H , . . . ,g nH } . Then \N : H\ < oo,
since N / H is finite as G / H is a periodic FC-group. Clearly g\,g2, ■■■ ,g n belong
to N and hence a = Y ^ = i ri9i e ^ ^
Thus I fl R N ^ 0.
We see that R N /R H is a finite normalizing extension of rings such that R N is
R H -free and both R N and R H are prime. Hence by [9, Theorem 4.16], we have
( / fl R N ) fl R H = I fl R H ^ 0. The result now follows from Lemma 2.1.
Groenewald [7, Theorem 2.1] proved that R G is strongly prime if R is strongly
prime and G is a unique product group. We shall need later the following.
Lemma 2.4. [7, Theorem 2.2] Let R be a ring and let H < G such that G / H is
a right ordered group. Then R G is strongly prime if R H is strongly prime.
3. One Relator Groups
One relator groups have been extensively studied recently. A group is said to be
locally indicable if each of its nontrivial finitely generated subgroups has the infinite
cyclic subgroup as homomorphic image. Howie [10, Corollary 4.3] proved that all
torsion free one relator groups are locally indicable. Burns and Hale [2, Corollary 2]
proved that locally indicable groups are right orderable ( R O ). Combining the two,
we get that every torsion free one relator group is an R O group. Also, i?0-groups
are unique product (UP) groups. This settles the question raised by Lichtman [12,
§4]. We refer to [6] and [13] as standard references for one relator groups.
The main result in this section is to prove the following.
Theorem 3.1. The group ring R G is strongly prime if R is strongly prime, G is
any one relator group with at least two generators.
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95
P ro o f. Let G = ( X ;S ) with |X| > 2 be a one relator group and R be any strongly
prime ring. If G is torsion free then it is an RO-group as noted above. Also, by
Lemma 2.4, R G is strongly prime.
Now, suppose that G is not torsion free. Then G = ( X ; S ) , S = T n, n > 2, T
is not a proper power and T is a cyclically reduced word. By [6 , Theorem 2] G
contains a torsion free normal subgroup Go with |G : Go I < 00 •
Let H be the subgroup of G generated by all torsion elements of G. Then by [6 ,
Theorem 1] H is the free product of all conjugates of the cyclic subgroup generated
by the image of T. Clearly G /H = (X -,T ) is a torsion free one relator group. Thus
G /H is an .RO-group.
Now Go H H is a torsion free subgroup of H which is a free product of finite
cyclic groups, hence by Kurosh subgroup Theorem [13, IV. 1.10], Go H H — F is a
free group.
Further G q /F = Gq/Gq fl H = G q H /H and G q H /H is a subgroup of an RO group G / H , hence G o /F is an .RO-group. Also F is an .RO-group, since it is free.
Thus Go is an .RO-group [14, 13.1.5] and RGq is strongly prime by Lemma 2.4.
Finally it is not difficult to see that the locally finite radical of G is trivial, that
is, L(G) = ( 1). Now by Lemma 2.3, R G is strongly prime, since R G q is strongly
prime and G /G o is finite.
4. FC-Solvable and F C -H ypercen tral G rou ps
Following Duguid and McLain [4], a group G is called FC-solvable if it has a
finite subnormal series,
1 = Ho <! H\
H 2 55 •••^ H n -i — H n — G
such that H i - 1 < Hi and H i /H i - i is an FG-group for i = 1 ,2 ,... , n. For an
arbitrary group G, the F C - centre A (G ) = {x € G | |G : Cg{x)\ < 00} and
the torsion part of A (G ) is A + (G) = {x € A (G ) | x has finite order}. A(G )
and A + (G) are characteristic subgroups of G, A ( G ) /A + (G) is torsion free abelian,
A + (G) is locally finite and A + (G) is the union of all finite normal subgroups of G
([14], Chapter 4).
Thus G is an FC-group if and only if G = A (G ). Also for any group L(G ) = ( 1)
implies A + (G) = ( 1) and A (G ) is torsion free abelian.
T h eorem 4.1. Let R be a ring and G an FC-solvable group. Then R G is strongly
prime if and only if R is strongly prime and L(G ) = (1).
P ro o f. If R G is strongly prime, then the conclusion follows from [8 , IILl.(a)].
Conversely suppose that R is strongly prime and L(G) = ( 1). Since G is F C solvable, it has a finite subnormal series 1 = Ho <! Hi < H 2 < . .. < Hn_ 1 < H n =
G such that H i - 1 < Hi and H i/H i - 1 is an FC-group for i = 1 ,2 ,... ,n.
We shall prove, by induction, that RHk is strongly prime for k = 1 ,2 ,... ,n.
First of all, H\ is subnormal in G and L(G) = (1), hence by Lemma 2.2, L(H\) =
(1). Also H\ is an FC-group, so Hi is torsion free abelian. Thus R H i is strongly
prime by [8 , IILl.(b)].
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R.K. SHARMA and J.B. SRIVASTAVA
Assume that RHi is strongly prime for i = 1 ,2 ,... ,k. Now H k+i / H k is an
FC-group, hence its torsion subgroup is given by ( H k+i / H k)+ = \[Hkf H k^ where
y/Jh = { z € H k+1 |x m e H k for some m > 1}. Clearly v'tffc < H k+ 1, s /W kf H k is
a periodic FC-group, and H k+\/y/TTk is torsion free abelian.
Further L(\fH k) — (1) by Lemma 2.2, since y/Hk is subnormal in G and L (G ) =
(1). Now R H k is strongly prime, \JTTk/Hk is a periodic FC-group, so by Lemma
2.3, R^/Hk is strongly prime.
Next Ry/Tlk is strongly prime, H k+i/y/TTk is torsion free abelian, hence ordered
and therefore by Lemma 2.4, R H k+\ is strongly prime. The induction is complete
and R G = R H n is strongly prime, as desired.
Corollary 4.2. Let R be a ring and G a nilpotent group. Then R G is strongly
prime if and only if R is strongly prime and G is torsion free.
Corollary 4.3. Let R be a ring and G a solvable by finite group.
strongly prime if and only if R is strongly prime and L(G ) = (1).
Then R G is
The FC'-chain of a group G is defined by A i(G ) = A (G ), Aa+i(G?)/Aa(G ') =
A ( G /A \ ( G ) ) for an ordinal A, and A fX(G) = Ua<mAa(G) if p is a limit ordinal.
The group G is called hyper-A or FC-hypercentral if G = A a(G) for some ordinal
a. It is easy to see that G is hyper-A if and only if A ( G / H) is nontrivial for every
proper normal subgroup H of G. The following theorem can be proved on the lines
similar to Theorem 4.1, using transfinite induction.
Theorem 4.4. Let 1 = Go < Gi <
< G a = G for some ordinal a be a
series of normal subgroups of G, where G^ = ^\<^G\ for a limit ordinal
and
G \ + i/G \ is an FC-group for every A. Then R G is strongly prime if and only if
R is strongly prime and L(G ) = (1).
Corollary 4.5. Let R be a ring and G a hyper-A group.
prime if and only if R is strongly prime and L(G ) = (1).
Then R G is strongly
Corollary 4.6. Let G be a group having a finite normal series 1 = Go <! G\ <
... < G n = G such that each factor G i /G i - i is hyper-A.
prime if and only if R is strongly prime and L (G ) = (1).
Then R G is strongly
Proof. Let R be strongly prime and L(G ) = (1). Then L(G\) = (1) by Lemma
2.2 and G\ is hyper-A, so RG\ is strongly prime by Corollary 4.5. Next assume
that RGi is strongly prime for some i > 1. Since Gi+ i /G i is hyper-A, there exists
a series Gi — Ho < H\ <
< H a = G l+\ for some ordinal a such that each
H\ < Gi + 1 and H\+ i/H \ is an FC-group. So we get that RG i + 1 is strongly
prime. Thus R G = R G n is strongly prime.
5. Group Algebras
In this section we take R = K to be a field. In [1] strongly prime group algebras
over fields are studied. Many authors in many contexts have obtained sporadic
results on strongly prime group algebras using known intersection theorems and
the existing related group ring results, (cf. [14]). We strictly adhere to certain
STRONGLY PRIME GROUP RINGS
97
comments, remarks and basically some new examples to show that the study o f
strongly prime group algebras is very much desirable.
First we observe that for a nilpotent group G, the group algebra K G is strongly
prime if and only if G is torsion free if and only if K G is a domain. However, if
G is a polycyclic by finite group then K G is Noetherian and so K G is strongly
prime if and only if K G is prime. When K G is strongly prime, at times K G is
a domain or very close to a domain, however sometimes it is just prime and far
from being a domain. We show that any group can be embedded in an infinite
simple group whose group algebra is strongly prime. Also we conclude via wreath
products that any infinite group can occur as a homomorphic image of a group
whose group algebra is strongly prime.
Proposition 5.1. Let K be a field and G a group. Then G can be embedded in
an infinite simple group G* such that K G * is strongly prime.
Proof. Let G\ = G x
where C 00 = (x) is infinite cyclic. By [14, 9.4.4] G\ can
be embedded into an algebraically closed group G* which is infinite simple. By
[14, 9.4.5, 9.4.6], the augmentation idal u>(KG*) is the only nonzero proper ideal
in K G * . Now lj(K G *) contains x — l and A nn^G * {x — 1) = 0, since x has infinite
order. Thus by definition K G * is strongly prime.
On the contrary by [14, 9.4.9, 9.4.10] if G is a universal locally finite group, then
G is infinite simple and the augmentation ideal uj(KG) is the only nonzero proper
ideal of K G , but K G is not strongly prime.
Our next result gives a rich source of examples.
Proposition 5.2. Let G — A \ B be the wreath product of groups A and B with
A 7^ (1) and B infinite.
If H = YlbeB Af, is the base group such that K H is
strongly prime, then K G is also strongly prime.
Proof. By [14, 9.2.7], I fl K H ^ 0 for any nonzero ideal I of K G . The result
follows by Lemma 2.1.
Remark 5.3. Let A be a non trivial torsion free solvable group and B be any
infinite group. Then K A is a domain by [11]. Thus if G = A \ B and K is a field,
then K G is strongly prime.
Acknowledgements. The authors wish to express their sincere thanks to the
referee for valuable suggestions.
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R.K. SHARMA and J.B. SRIVASTAVA
98
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R .K . Sharma
J.B. Srivastava
Indian Institute of Technology
Indian Institute of Technology
Kharagpur 721302
New Delhi 110016
IN D IA
IN D IA