KYUNO, SHOJI

Osaka City University Repository

Kyuno, S.
Osaka J. Math.
12 (1975), 639-645
ON THE RADICALS OF Γ-RINGS
SHOJI KYUNO
(Received October 17, 1974)
1. Introduction
N. Nobusawa [1] introduced the notion of a Γ-ring, more general than a
ring, and proved analogues of the Wedderburn-Artin theorems for simple Γ-
rings and for semi-simple Γ-rings Barnes [2] obtained analogues of the classical
Noether-Lasker theorems concerning primary representations of ideals for Γ-
rings; Luh [3,4] gave a generalization of the Jacobson structure theorem for
primitive Γ-rings having minimum one-sided ideals, and obtained several other
structure theorems for simple Γ-rings; Coppage-Luh [5] introduced the notions
of Jacobson radical, Levitzki nil radical, nil radical and strongly nilpotent radical
for Γ-rings and Barnes' [2] prime radical was studied further. Also, inclusion
relations for these radicals were obtained, and it was shown that the radicals all
coincide in the case of a Γ-ring which satisfies the descending chain condition
on one-sided ideals.
In this paper the notions of semi-prime ideals are extended to Γ-rings, and
it is shown that all of the following conditions are equivalent: (1) Q is a semi-
prime ideal. (2) Qc is an ra-system. (3) The Γ-residue class ring MjQ contains
no non-zero strongly nilpotent ideals. (4) The prime radical P(Q) of the ideal
Q coincides with Q. Also, the following characterization of P(M) is obtained.
P(M) is a semi-prime ideal which is contained in every semi-prime ideal in M.
Let R be the right operator ring of a Γ-ring M. For P^R and for Q^M we
define P*= {x^M: [Γ, x]^P} and Q*'= βΓJfα,, x,]eR:MζΣ[ai9 *,])c0}.
In [5] the following theorem was proved. If P(M) is the prime radical of the
right operator ring R of the Γ-ring M, then P(M)=P(JR)*.
We show the following result dual to the above theorem, P(R)=P(M)*'.
As a result, it is obtained that P(M)*'*=P(M) and P(R)**'=P(R). The similar
properties hold for the Levitzki nil radical and Jacobson radical. Also, some
radical properties are cosidered.
2. Preliminaries
Let M and Γ be additive abelian groups. If for all a,b,c^M, and
640 S. KYUNO
the following conditions are satisfied, (1) aab^M (2) a(a+β)b=aab+aβb,
aa(b+c)=aab+aac, (a+b)ac=aac+bac (3) (aab)βc=aa(bβc)y then following
Barnes [2], Mis called a Γ-ring. If these conditions are strengthened to, (!')
aab^M, aaβ^T (2'} same as (2) (3') (aab)βc=a(abβ)c=aa(bβc) (4) xγy=Q
for all x, y eM implies 7—0, then M is called a Γ-ring in the sense of Nobusawa
[1]. If A and B are subsets of a Γ-ring M and Θ^Γ, we denote AΘB, the
subset of M consisting of all finite sums of the form Σ#t αA > where a^A,
i
bf^B, and <2, eθ. For singleton subsets we abbreviate this notation, for
example, {a}ΘB=aθB. A right (left) ideal of a Γ-ring M is an additive sub-
group I of M such that ITM^I (MΓ/e/). If / is both a right ideal and a left
ideal, then we say that / is an ideal, or a two-sided ideal of M. For each a of a
Γ-ring M, the smallest right ideal containing a is called the principal right ideal
generated by a and is denoted by |α>. Similarly we define <a\ and <α>, the
principal left and two-sided (respectively) ideals generated by a. Let / be an
ideal of a Γ-ring M. If for each a+I, b-\-I in the factor group M//, and each
γeΓ, we define (a+I)rγ(b+I)=alγb+I9 then M// is a Γ-ring which we shall
call the Γ-residue class ring of M with respect to /. Let M be a Γ-ring and F
the free abelian group generated by Γ X M. Then
A = {Σ%(γ, , xf)^F: αeM=Φ Σ^WΛ = 0}
is a subgroup of F. Let R=F/A, the factor group, and denote the coset (γ, #)
+^4 by [γ, #]. It can be verified easily that [α, #] + [α, y]=[#> #+j] and [α, tf]
+ [/?, x]=[a+β9 x] for all ct, /5eΓ and Λ^eM. We define a multiplication in
Λ b y
If we define a composition on MxR into M by «
I I
then M is a right Λ-module, and we call I? the right
t
operator ring of the Γ-ring M. Similarly we may construct a left operator ring
L of M so that M is a left L-module. If A is a right (left) ideal of R (L), then
MA (AM) is an ideal of M. For subsets N^M, Φ^Γ, we denote by [Φ, N]
the set of all finite sums Σh^ χϊ\ m ^> where γt eΦ, xf^N, and we denote by
[(Φ, JV)] the set of all elements [φ, x] in Ry where φ^Φ, x^N. Thus, in
particular, R=[Γ, M]. For P^R we define P*= {a<=M: [Γ, α] = [Γ, {a}]^P}.
It then follows that if P is a right (left) ideal of R, then P* is a right (left) ideal
of M. Also, for any collection C of sets in R, Γ(P*= (ΠP)*. For ρ^M we
Pe£ Pe£
define ρ*'= β] [^ , Λ?J e Λ : M(Σ [«ί, ^  ]) ^  8} Then it follows that if Q is at t
right (left) ideal of M, then Q*' is a right (left) ideal of R. Also for any collec-
RADICALS OF Γ-RINGS 641
tion 3) of sets in M, ΓΊ £?*'=( Π £?)*'. For other notions relevant to Γ-rings we
βe.fi) <2e.0
refer to [5],
3. Semi-primeness
Following Barnes [2] an ideal P of a Γ-ring M is prime if for any ideals
A, B^M, ATB^P implies A^P or B^P. A subset S of M is an Trc-system
in M if 5=φ or if ay b<=S implies <#>Γ<Z>> Π S Φ φ. Barnes [2] has shown that
an ideal P is prime if and only if its complement Pc is an m-system. The
prime radical P(A) of the ideal A in a Γ-ring M is the set consisting of those
elements r of M with the property that every wz-system in M which contains r
meets ^4 (that is, has nonempty intersection with A). The prime radical of the
zero ideal in a Γ-ring M may be called the prime radical of the Γ-ring M which
we denote by P(M). Barnes [2] has characterized P(M) as the intersection of
all prime ideals of M. We now make the following definition. An ideal Q in a
Γ-ring M is said to be a semi-prime ideal if and only if it has the following
property: If A is an ideal in M such that AΓA^Q, then A^Q. It is clear
that a prime ideal is semi-prime. Moreover, the intersection of any set of semi-
prime ideals is a semi-prime ideal. It follows easily by induction that if Q is a
semi-prime ideal and A is an ideal and (AΓ)MA=(AΓAΓ—AΓ)A^Q for an
arbitary positive integer ny Aς^Q. Following Coppage-Luh [5] a subset S of M
is strongly nilpotent if there exists a positive integer n such that (ST)*S=Q. We
state the following theorem whose proof we omit since it can be established by
very easy modifications of the proof of Theorem 4.11 in [6].
Theorem 3.1. An ideal Q in a T-ring M is a semi-prime ideal in M if and
only if the T-residue class ring M/Q contains no nonzero strongly nilpotent ideals.
The following result is easy to prove.
Theorem 3.2. If Q is an ideal in a Γ-ring M, the following conditions are
equivalent: (1) Q is a semi-prime ideal. (2) If a^M such that <X>Γ<X>c;£),
thena^Q.
A set N of elements of a Γ-ring M is said to be an //-system if N=φ or if
a^N implies <fl>Γ<X>n-/VΦφ. The equivalence of conditions (1) and (2) of
Theorem 3.2 assures us that an ideal Q in a Γ-ring M is semi-prime if and
only if its complement Qc is an w-system. By proofs analogous to Lemma 4.14
and Theorem 4.15 in McCoy [6] we have the following results.
Lemma 3.3. If N is an n-system in a T-ring M and a^N, there exists an
m-system L such that
Theorem 3.4. An ideal Q in a T-ring M is a semi-prime ideal in M if and
642 S. KYUNO
only if P(Q)=Q.
In view of Barnes' characterization of P(M) as the intersection of all prime
ideals of M (Theorem 7 in [2]) we have the following immediate corollary to the
preceding theorem.
Corollary 3.5. If Q is an ideal in a Y-rίng M, then P(Q) is the smallest semi-
prime ideal in M which contains Q.
We have the following characterization of P(M) which follows immediately
from Corollary 3.5 and Theorem 7 in [2].
Theorem 3.6. P(M) is a semi-prime ideal which is contained in every semi-
prime ideal in M.
4. The prime radical
Coppage-Luh [5] have proved the following Lemma 4.1 and Theorem 4.2.
Lemma 4.1. If P is a prime ideal of R, then P* is a prime ideal of M.
Theorem 4.2. If P(R) is the prime radical of the right operator ring R of the
T-ring M, then P(M)=P(R)*.
We prepare the following lemma.
Lemma 4.3. If Q is a prime ideal of M, then Q*' is a prime ideal of R.
This proof is found in the proof of Theorem 4.1 in [5]. We now prove the
following theorem dual to Theorem 4.2.
Theorem 4.4. If P(R) is the prime radical of the right operator ring R, then
P(R)=P(M)*'.
Proof. Let P be a prime ideal of R, by Lemma 4.1 P* is a prime ideal of
M. Let us set P*=£). Then by Lemma 4.3 Q*' is a prime ideal of R. Since
Q*'={r(ΞR: Mr^Q} = {r<=R: [Γ, MrjcP}, it follows that RQ*'=[Γ, M]Q*'
= [Γ, M£>*']cP. Thus by the primeness of P, £>*'^ P. Also, [Γ, MP]=
[Γ,M]P=RP^P. Hence, Pcρ*'. Therefore P=Q*'. It follows that P(P),
which is the intersection of all prime ideals of P, contains (Ί (?*'=( Π £?)*'>
Q(Ξ<£) Q(Ξ£)
where 3) is a certain collection of prime ideals of M. But (Π Q)*'^P(M)*', so
Q^S)
we conclude that P(/ZpP(A/)*'. On the other hand, P(M)*'=( (Ί £>)*'= IΊ Q*',
where the intersection is taken over all prime ideals of M. Since, by Lemma 4.3
each Q*' is a prime ideal of R, and P(R) is the intersection of all prime ideals
of R, it follows P(R) eP(M)*'. Thus P(P)=
RADICALS OF Γ-RINGS 643
The following result is a consequence of Theorem 4.2 and Theorem 4.4.
Theorem 4.5. If P(K) is the prime radical of the right operator ring R, then
P(M)=P(M)*'* mdP(R)=P(R)**'.
The next theorem follows immediately from previous Theorem 4.4 and
Theorem 4.2 in [5].
Theorem 4.6. // / is an ideal of a Γ-ring M, then P(I)*'=I*' Π P(R),
where P(I) denotes the prime radical of I considered as a T-ring.
5. The Levitzki nil radical
Following Coppage-Luh [5] a subset S of a Γ-ring M is locally nilpotent if
for any finite set F^S and any finite set Φ<ΞΓ, there exists a positive integer n
such that (FΦ)nF=Q. Also the Levitzki nil radical of M is the sum of all locally
nilpotent ideals of M and is denoted by X(M).
Coppage-Luh [5] have proved the next theorem.
Theorem 5.1. If M is a T-ring then X(M)=j:(R)*, where -C(R) is the
Levitzki nil radical of the right operator ring R of M.
We know the following result whose proof will be found in Jacobson [7],
p. 163.
Theorem 5.2. -C(R) is a locally nilpotent ideal.
We prove the following two lemmas.
Lemma 5.3, If J is a locally nilpotent ideal of a T-ring M, then /*' is a
locally nilpotent ideal of R, where R is the right operator ring of M.
Proof. A finite subset of /*' is a subset of [φl9 F^, where φ1 is a finite subset
of Γ and Fl is a finite subset of M. Since M[φly FJc/, MφJF^J. Thus
FiφJ?! is a finite subset of/. Since/ is locally nilpotent, (F1φ1F1φ1)nF1φ1F1=0
for some n. Hence, (F.φ^^F^O. Thus [φ
n
 Ftf*+*=[φl9 (Flφ1)2li+1ίiJ=
[φlf 0]=0. Then /*' is locally nilpotent.
Lemma 5.4. // M is a T-ring then -C(R)<^£(M)*', where -C(R) is the
Levitzki nil radical of the right operator ring R of M.
Proof. Since _£*(/?) is an ideal by Theorem 5.2., we have
so [Γ, M]£(R) c £(R), that is, [Γ, MX(R)} c £(R). Thus, MX(R) c -C(R)*. By
Theorem 5.1., we get MX(R)^X(M). Therefore J7(/Z)c
Theorem 5.5. // M is a T-ring then £(M)*'=£(R\ where X(R) is the
644 S. KYUNO
Levitzki nil radical of the right operator ring R of M.
Proof. In view of Lemma 5.3 and the fact that X(M) is a locally nilpotent
ideal (Theorem 7.1 in [5]), we know that J2(M)*' is a locally nilpotent ideal of
R. By the definition of £(R), J?(M)*'c £(R). On the other hand by Lemma
5.4, .£(Λ)c_£(M)*'. Hence £(M)*'=£(R).
The next theorem follows from Theorem 5.1 and Theorem 5.5.
Theorem 5.6. If -C(R) is the Levitzki nil radical of the right operator ring
Rofa T-ring M, then £(M)*'*=£(M) and £(R)**'=-C(R).
The next theorem follows immediately from previous Theorem 5.5 and
Theorem 7.3 in [5].
Theorem 5.7. // / is an ideal of a T-ring M, then £(!)*' '=/*' ΓΊ -C(R).
6. The Jacobson radical
Following Copppage-Luh [5] an element a of a Γ-ring M is right quasi-
regular (abbreviated rqr) if, for any γeΓ, there exists ι?/eΓ, #,eM, *=1,2, ••-,»
such that
jiXi— Σ xγwiiXi = 0 for all
ί = 1 » = 1
A subset S of M is rqr if every element in S is rqr. £(M)= {a^M: <#> is
rqr} is the right Jacobson radical of M. Copppage-Luh [5] have shown the
following Lemma 6.1 and Theorem 6.2.
Lemma 6.1. An element a of a T-ring M is rqr if and only if, for all 7 e Γ,
[γ, α] is r#r m the right operator ring R of M.
Theorem 6.2. If M is a T-ring then g(M)=/(R)*, where £(R) denotes the
Jacobson radical of the right operator ring R of M.
We prove the following theorem dual to Theorem 6.2.
Theorem 6.3. // M is a T-ring, then g(R)=g(M)*'.
Proof. If Σ [γ, , *J <EE ^ (Λf)*7, then for all X<EΞ M
*(
ί
By Theorem 6.2, for all
[Ύ, Σ*TΛ] = [Ύ,
Hence R Σ [T, , ^ ί] is rqr. By the difinition of £(R)> Σ [O'ί, «J e ^ (Λ) and then
t I
RADICALS OF Γ-RINGS 645
If Σ[S, , xf](Ξ/(R), for all -y<ΞΓ and all
I
[7, *]Σ[S, , x<] = [Ύ,
Thus
- {[7, 0]
Hence, [7, a] is rgr for all 7 e Γ and for <z e <Σ tf8t-tf,->. By Lemma 6.1
<Σ*δ,*,> is r#r and then #(Σ[δί, Λ
ί
])=Σ^S
ί
Λ?/e^(Λί), that is, M(Σ[δ, , ^ , ])^
/(M). Therefore Σ Pi, Λ?J e ^ (M)#/. Then /(#) c^(Af )*'. Thus the proof
is completed.
By Theorem 6.2 and Theorem 6.3 we have the following theorem.
Theorem 6.4. ^(M)=/(M)*'* and g(R)=g(R)**'.
The next result follows from Theorem 6.3 and Theorem 8.4 in [5].
Theorem 6.5. If I is an ideal in a T-ring M, $(!)*' =1*'
TOHOKU GAKUIN UNIVERSITY
References
[1] N. Nobusawa: On a generalization of the ring theory, Osaka J. Math. 1 (1964), 81-
89.
[2] W.E. Barnes: On the Γ-rings of Nobusawa, Pacific J. Math. 18 (1966), 411-422.
[3] J. Luh: On primitive Γ-rings with minimal one-sided ideals, Osaka J. Math. 5
(1968) 165-173.
[4] J. Luh: On the theory of simple Γ-rings, Michigan Math. J. 16 (1969), 65-75.
[5] W.E. Coppage and J. Luh: Radicals of gamma rings, J. Math. Soc. Japan 23 (1971),
40-52.
[6] N.H. McCoy: The Theory of Rings, Macmillan, N.Y. 1964.
[7] N. Jacobson: Structure of Rings, revised ed., Amer. Math. Soc. Colloquim Publ.
37, Providence, 1964.



On the radicals of Γ -rings

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