Strongly prime gamma rings and Morita equivalence of rings

Abstract

We define β-insulator and strongly prime Γ-rings. We also prove the following main results: (i) If a Γ-ring M is weakly semiprime then M is strongly prime if and only if its left operator ring L is right strongly prime and its right operator ring R is left strongly prime. (ii) If M is strongly prime Γ-ring, then its left and right operator rings are Morita equivalent.

Full text

Southeast Asian
Bulletin of
Mathematics
c
°SEAMS. 2008
Southeast Asian Bulletin of Mathematics (2008) 32: 1137–1147
Strongly Prime Gamma Rings and Morita Equivalence
Of Rings
C. Selvaraj∗and S. Petchimuthu
Department of Mathematics, Periyar University, Salem - 636 011, India.
Email: selvavlr@yahoo.com
R. George
Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli
627012, India.
AMS Mathematics Subject Classification(2000): 16S60, 20M12, 16Y99
Abstract. In this paper, we define β-insulator and strongly prime Γ-rings. We also
prove that the following main results:
(i) If a Γ-ring Mis weakly semiprime then Mis strongly prime if and only if its
left operator ring Lis right strongly prime and its right operator ring Ris left
strongly prime.
(ii) If Mis strongly prime Γ-ring,then their left and right operator rings are Morita
equivalent.
Keywords: Semisimple; Semiprime; Insulator; Morita equivalence.
1. Introduction
N. Nobusawa [8] introduced the notion of a Γ-ring, which is more general than a
ring. Later on, W.E. Barnes [2] weakened slightly the conditions in the definition
of a Γ−ring in the sense of Nobusawa. Actually, W.E. Barnes [2], J. Luh [7],
and S. Kyuno [6] studied the structures of Γ-rings and obtained various gener-
alizations analogous to the corresponding parts in ring theory. Strongly prime
rings were then introduced by Handelmann and Lawrence [4]. In this paper, we
extend the concept of strongly primeness to Γ-rings.
2. Preliminaries
In this section, we recall certain definitions and results needed for our purpose.
Received June 13 2006, Accepted May 15 2007.
∗Research supported by University Grants Commission, India

1138 C. Selvaraj et al.
Definition 2.1. Let Mand Γbe additive abelian groups. If for all a, b, c ∈M
and α, β ∈Γ, the following conditions are satisfied
(i) aαb ∈M,
(ii) (a+b)αc =aαc +bαc,a(α+β)c=aαc +aβc,aα (b+c) = aαb +aαc,
(iii) (aαb)βc =aα (bβc),
then Mis called a Γ-ring. If these conditions are strengthened to
(i0)aαb ∈M, αaβ ∈Γ,
(ii0)same as (ii),
(iii0)(aαb)βc =a(αbβ)c=aα (bβc),
(iv0)aγb = 0 for all a, b ∈Mimplies γ= 0,
then Mis called a Γ-ring in the sense of Nobusawa [8].
Definition 2.2. A right (left) ideal of a Γ−ring Mis additive subgroup of a Γ-
ring Msuch that IΓM⊆I(MΓI⊆I).If Iis both a right and a left ideal,
then we say that Iis an ideal of M. An ideal Iof a Γ-ring Mis said to be
prime(semiprime) if for any ideals U, V ⊆M, U ΓV⊆Iimplies U⊆Ior
V⊆I(UΓU⊆I⇒U⊆I). If a∈M, then the principal ideal generated by
a, denoted by < a >, is the intersection of all ideal containing aand is the set
of all finite sum of the form na +xαa +aβy +uγaδv where nis an integer, a,
x, y, u, v are elements of Mand α, β, γ, δ are elements of Γ.
Definition 2.3. Let Mbe a Γ-ring. Consider the maps [α, x] : y7→ yαx and
[x, α] : y7→ xαy, x ∈M , α ∈Γand for all y∈M. Clearly [x, α],[α, x]belongs
to EndM . The bilinearity map Γ×M→EndM (M×Γ→EndM )given by
(α, m)7→ [α, m] ((m, α)7→ [m, α]) gives rise to a linear map from Γ⊗ZM7→
EndM (M⊗ZΓ7→ EndM)given by Piαi⊗mi7→ Pi[αi, mi](Pimi⊗αi7→
Pi[mi, αi]) , αi∈Γand mi∈M. The image of Γ⊗ZM(M⊗ZΓ) in EndM is
an associative ring denoted by R(L)and call it right(left) operator ring of M.
The ring multiplication in Rand Lis given by
X
i
[αi, xi]X
j
[βj, yj] = X
i,j
[αi, xiβjyj]
and X
i
[xi, αi]X
j
[yj, βj] = X
i,j
[xiαiyj, βj].
Moreover, Mis faithful L−Rbimodule, where if x∈M,
r=X
t
[δt, zt]∈Rand `=X
i
[xi, αi]∈L
then xr =Ptxδtztand `x =Pixiαix.

Strongly Prime Gamma Rings and Morita Equivalence of Rings 1139
Definition 2.4. AΓ-ring Mis said to be left(right) weakly semiprime Γ−ring if
[x, Γ] 6= 0 ([Γ, x]6= 0) for all x6= 0 ∈M.
Mis said to be weakly semiprime if it is both left and right weakly semiprime.
Definition 2.5. Let Mbe a Γ−ring, an element m∈Mis said to be left zero
divisor if mαn = 0 for some α∈Γimplies that n6= 0. An element nis said to
be right zero divisor if mαn = 0 implies that m6= 0. An element in a Γ-ring M
is said to be zero divisor if it is both left and right zero divisor.
Definition 2.6. [4] A ring Ris said to be prime if for given x6= 0, y 6= 0 ∈R,
then there exists z∈Rsuch that xzy 6= 0.
We recall that annihilator of a subset Aof a ring Xis Ann(A) = {r∈
X/Ar = (0)}.
Definition 2.7. [4] A right insulator for x6= 0 ∈Rto be a finite subset of R, S (x)
such that Ann ({xy/y ∈S(x)}) = (0) .
Then, Ris said to be strongly prime if each non zero element of Rhas a
right insulator. That is, for each x6= 0,there is a finite set S(x)such that for
y∈R, {xzy/z ∈S(x)}= (0) implies y= 0.
Definition 2.8. [1] Let Sand Tbe arbitrary associative rings with unity. By
Mod −T(T−Mod)we denote the category of all right(left) T-modules. Then
a module Mis said to be a generator (in Mod-T)if for every T-module K
there is a set Isuch that the sequence MI→K→0is exact. We call Ma
progenerator if it is finitely generated, projective and is a generator. The rings S
and Tare said to be Morita equivalent if S-Mod (Mod-S) and T-Mod (Mod-T)
are equivalent categories. Equivalently Sand Tare Mortia equivalent if there
exists a progenerator MTwith S∼
=EndT(M).
Theorem 2.9. [10] Let Mbe a weakly semiprime Γ-ring, Land Rbe its operator
rings. Then Land Rare Morita Equivalent.
3. Prime and Semiprime Ideals of Γ-Rings
In this section, we shall give the basic connection between prime ideals and
semiprime ideals of Γ-rings.
Definition 3.1. [6] An ideal Qin a Γ-ring Mis said to be semiprime ideal if for
any ideal Uof M, U ΓU⊆Qimplies U⊆Q.
Definition 3.2. [6] A subset Sof a Γ-ring Mis said to be an m-system if S=φ

1140 C. Selvaraj et al.
or if a, b ∈Simplies < a > Γ< b > ∩S6=φ.
Definition 3.3. [2] For any ideal Uof a Γ-ring, we define m(U)to be the set of
all elements xof Msuch that every m-system containing xcontains an element
of U.
Definition 3.4. A subset Nof a Γ-ring Mis said to be an n-system if N=φor
if a∈Nimplies < a > Γ< a > ∩N6=φ.
Lemma 3.5. Let Mbe a Γ−ring. Then, an ideal Qin Mis semiprime if and
only if QC.is an n-system.
Proof. Suppose that Qis a semiprime ideal and let a∈QC,then a /∈Q. Since
Qis semiprime, it follows from [6, Theorem 1] that <a>Γ<a>6⊂ Q. This
implies that < a > Γ< a > ∩QC6=φ, so that QCis an n−system.
Conversely, suppose QCis an n-system and let a6∈ Q. We shall prove that
< a > Γ< a >6⊂ Q. Since QCis an n−system, < a > Γ< a > ∩QC6=φ.
Take z∈<a>Γ< a > ∩QCso that z∈< a > Γ<a>and z /∈Q. Hence
< a > Γ< a >6⊂ Q. Thus Qis a semiprime ideal.
Definition 3.6. For any ideal Uof a Γ-ring M, we define n(U)to be the set of
all elements xof Msuch that every n−system containing xcontains an element
of U.
Lemma 3.7. Let Mbe a Γ−ring in the sense of Nobusawa and N⊆Man n-
system, If Pis an ideal maximal with respect to the property that Pis disjoint
from N, then Pis a semiprime ideal.
Proof. Suppose that < a > Γ< a >⊂Pand a /∈P. Then, by the maximal
property of P, there exists x∈Nsuch that x∈P+< a > . Since Nis a an
n−system, < x > Γ< x > ∩N6=φ. Let z∈< x > Γ< x > ∩N. Then zis the
finite sum of elements of the form
(nx +cαx +xβd +eγxδf )ρ(mx +gµx +xνh +jξxηk),
where mand nare integers, c, d, e, f, g, h, j, x and kare in Mand α, β, δ, ρ, µ,
γ, ξ , η, ν in Γ.But every element in such a product is in Pby condition (i0),(iii0)
of Definition 2.1 and the assumption that < a > Γ< a >⊂P.
For example,
(cαx)ρ(gµx) = cα (xρ (gµx)) = cα (xρ (gµx)) = cα (x(ρgµ)x)
∈cα (xΓMΓx)⊆cα [(P+< a >) ΓMΓ (P+< a >)]
⊆cα [P+< a > ΓMΓ< a >]
⊆P.

Strongly Prime Gamma Rings and Morita Equivalence of Rings 1141
Hence z∈P, a contradiction. Thus Pmust be a semiprime ideal.
Lemma 3.8. Let Mbe a Γ−ring in the sense of Nobusawa. If Uis any ideal in
M, then n(U)is equal to the intersection of all semiprime ideal containing U.
In particular, n(U)is an ideal in M.
Proof. We first prove the inclusion ‘ ⊆’. Let x∈n(U) and Pbe any semiprime
ideal containing U. Since Pis semiprime ideal, PCis an n-system. This n-system
can not contain x, for otherwise it meets Uand hence also P. Therefore, we have
x∈P.
Conversely, suppose that xbelongs to the intersection of all semiprime ideals
containing U. We show that x∈n(U).If x /∈n(U),then by definition there
exists an n−system Ncontaining xwhich is disjoint from U. By Zorn’s lemma,
there exists an ideal Pcontaining Uwhich is maximal with respect to being
disjoint from N. By Lemma 3.7, Pis a semiprime ideal and we have x /∈P,
which is a contradiction and hence x∈n(U).
Next we need the following lemma relating m-systems and n-systems.
Lemma 3.9. Let Sbe an m−system in a Γ−ring Mand let a∈S.Then there
exists an n−system N⊆Ssuch that a∈N.
Proof. We define N={a1, a2, . . .}inductively as follows: a1=a, since Sis an
m-system, let a2∈< a1>Γ< a1>∩S, then a2is the finite sums of the form
(n1a1+c1α1a1+a1β1d1+e1γ1a1δ1f1)ρ(m1a1+g1µ1a1+a1ν1h1+j1ξ1a1η1k1)
where a1, c1, d1, e1, f1, g1, h1, j1, k1are elements in Mand m1, n1are integers,
α1, β1, γ1, δ1, µ1, ν1, ξ1, η1are elements in Γ.Again, regard Sas an m−system,
take a3∈< a2>Γ< a2>∩S. We continue the similar fashion, we have the
elements a3, a4, . . . of N. Now for any i, < ai>Γ< ai>contains ai+1,an
element of N. Hence, < ai>Γ< ai>∩N6=φand N⊆Ssuch that a∈N.
Definition 3.10. [2] An ideal Qin a Γ−ring Mis said to be right primary if for
any ideal Uand V, U ΓV⊆Qimplies U⊆m(Q)or V⊆Q.
Theorem 3.11.Let Mbe a Γ-ring in the sense of Nebusawa. For any right pri-
mary ideal Qin M, the following statements are equivalent:
(i) Qis a prime ideal;
(ii) Q=n(Q) ;
(iii) Qis a semiprime ideal.
Proof. (i) ⇒(ii): Let Qbe a prime ideal, then Q⊆n(Q) is obvious. On the
other hand, let x∈n(Q) and suppose that x /∈Q. Since Qis prime, QCis an

1142 C. Selvaraj et al.
m-system by [2], and x∈QC. By Lemma 3.9, there exists an n-system N⊆QC
such that x∈N. But Nis disjoint from Q, therefore x /∈n(Q),which is a
contradiction. Hence x∈Q, so that n(Q)⊆Q.
(ii) ⇒(iii) is obvious.
(iii) ⇒(i): Suppose that Qis a semiprime ideal. We have to prove that Q
is a prime ideal. Let Uand Vbe any ideal in Mwith UΓV⊆Q. Since Qis
primary, UΓV⊆Qimplies that U⊆m(Q) or V⊆Q. Since Qis a semiprime
ideal, Q=m(Q)(see, [6]). Hence U⊆Qor V⊆Q. Thus Qis a prime ideal in
M.
Theorem 3.12. For any ideal Qin M,Qis prime if and only if Qis primary
and semiprime.
Proof. Suppose that Qis a prime ideal. We have to prove that Qis primary.
Let Uand Vbe any ideal in Msuch that UΓV⊆Q. Since Qis a prime ideal,
U⊆n(Q) or V⊆Qby Theorem 3.11. Now we claim is that n(Q)⊆m(Q).
Let x∈n(Q) and Sbe any m-system containing x. Since any m−system is an
n−system, Sis an n-system containing x. Since x∈n(Q), S meets Q. Hence
x∈m(Q) and therefore U⊆n(Q) or V⊆Qimplies that U⊆m(Q) or V⊆Q.
Hence Qis a primary ideal. Since every prime ideal is a semiprime ideal, Qis
semiprime. Thus Qis semiprime and hence primary ideal.
Conversely, suppose that Qis primary and semiprime ideal. By Theo-
rem 3.11, Qis a prime ideal.
4. Semiprime Γ-Rings
In this section, we shall the relate semiprime Γ-rings to semisimple Γ-rings.
Definition 4.1. Let Mbe a Γ−ring. Then, we call Msemiprime if (0) is a
semiprime ideal. Mis said to be prime if (0) is a prime ideal.
Definition 4.2. [8] Let Mbe a Γ-ring. If for any non zero element aof Mthere
exists an element γ(depending on a) in Γsuch that aγa 6= 0,we say that Mis
semisimple. If for any non zero element aand bof M, there exists γ(depending
on aand b) in Γsuch that aγb 6= 0,we say that Mis simple.
Theorem 4.3. Let Mbe a Γ−ring in the sense of Nobusawa. Then Mis semisim-
ple if and only if Mis semiprime.
Proof. Suppose that <a>Γ<a>= 0 for any a∈M. Since aΓa⊆<a>
Γ< a >, aΓa= 0.Since Mis semisimple, aΓa= 0 implies that a= 0. Hence
< a >= 0,and so Mis semiprime [6, Theorem 1].
Conversely, suppose aΓa= 0 for any a∈M. Since aΓMΓa⊆aΓa, aΓMΓa=
0. Since Mis semiprime, it follows from [6, Theorem 1] that a= 0. Hence Mis

Strongly Prime Gamma Rings and Morita Equivalence of Rings 1143
semisimple.
Corollary 4.4. Mis semiprime if and only if for any ideal U, V in M,UΓV= 0
implies that U∩V= 0.
Proof. Suppose that Mis semiprime. Let U, V be ideals in Msuch that UΓV=
0 and let x∈U∩V. Since xΓx⊆UΓV, xΓx= 0. Since Mis semiprime, Mis
semisimple by Theorem 4.3. Hence xΓx= 0 implies that x= 0 and consequently
U∩V= 0.
Conversely, suppose UΓU= 0 implies U∩U= 0 by hypothesis. Hence U= 0,
so that Mis semiprime.
5. Strongly Prime Γ-Rings
In this section, we shall prove that left(right) operator ring of a right(left)strongly
prime Γ-ring is right(left)strongly prime and also we shall prove that if Mis
a strongly prime Γ-rings then their left and right operator rings are Morita
equivalent.
Definition 5.1. Let Mbe a Γ-ring. If Ais a subset of M, then we define a
right (left) α-annihilator of Ato be a right(left) ideal rα(A) = {m∈M/Aαm =
0}(`α(A) = {m∈M/mαA = 0}).
We adopt the symbol M∗to denote the non zero element of M.
Definition 5.2. A right(left) β-insulator for a∈M∗is a finite subset of M,
Sβ(a), such that rα({aβc/c ∈Sβ(a)}) = (0) (`α({cβa/c ∈Sβ(a)}) = (0)),∀α∈
Γ.
Definition 5.3. AΓ-ring Mis said to be right (left) strongly prime if for every
β∈Γ,each non zero element of Mhas a right (left) β−insulator, that is for
every β∈Γand a∈M∗,there is a finite subset Sβ(a)such that for b∈M,
{aβcαb/c ∈Sβ(a)}= 0 ({bαcβa/c ∈Sβ(a)}= 0),∀α∈Γ implies b= 0.
AΓ-ring Mis said to be strongly prime if it is both left and right strongly prime.
Theorem 5.4. Let Mbe a Γ−ring with DCC on annihilators. Then Mis prime
if and only if Mis strongly prime.
Proof. Suppose that Mis right strongly prime. To prove Mis prime, let a, b ∈M
such that a6= 0 and b6= 0.Since Mis right strongly prime, for every β∈Γ, there
exists a right β−insulator Sβ(a) for a. Then rα({aβc/c ∈Sβ(a)}) = 0,∀α, β ∈

1144 C. Selvaraj et al.
Γ. Since b6= 0, b /∈rα({aβc/c ∈Sβ(a)}),∀α, β ∈Γ,there exists α, β ∈Γ,such
that aβcαb 6= 0 where c∈Sβ(a).Hence Mis prime.
Conversely, suppose that Mis prime. We have to prove that Mis right
strongly prime. Let m∈M∗and consider the collection of right α-annihilator
ideals of the form rα({mβn/n ∈I}),∀α, β ∈Γ where Iruns over all finite
subsets of Mcontaining the identity. Since Msatisfies the DCC on right an-
nihilators, choose a minimal element K. If K6={0},we can find an element
a∈Ksuch that a6= 0.Since Mis a prime Γ-ring, it follows from [6, Theorem
4] that there exists b∈M, such that mγbδa 6= 0 for γ , δ ∈Γ.
Let I0be a finite subset of Mcontaining the identity and b. Since mγbδa 6=
0, a /∈rδ({mβn/n ∈I0}),a contradiction. This forces that K={0}.Thus m
has a right β−insulator ∀β∈Γ. Since m∈M∗is arbitrary, every element of
M∗has a right β-insulator for all β∈Γ. Similarly every element of M∗has a
left β- insulator for all β∈Γ. Hence Mis a strongly prime Γ-ring.
Definition 5.5. Let Mbe a Γ-ring. Then a left ideal Iof Mis said to be essential
if I∩J6= 0 for all non zero left ideals Jof M.
Definition 5.6. The singular ideal of a Γ-ring Mis the ideal composed of elements
whose right α-annihilators for each α∈Γis an essential right ideal.
Theorem 5.7. If Mis a strongly prime Γ-ring having no zero devisor, then
singular ideal is zero.
Proof. Let Mbe a strongly prime Γ-ring and Abe a singular ideal. Suppose
that there exists an element a∈Asuch that a6= 0.Let Sβ(a) be a right
β−insulator for a. Since Ais an ideal, aβb ∈A, ∀b∈Sβ(a).Now rα({aβb}) =
{x∈M/ (aβb)αx = 0}implies that aβbαx = 0,∀x∈rα(aβb), b ∈Sβ(a).
Then aβbαrα({aβb})=0.Hence aβbα [∩rα({aβb})] = 0.Since Ais singular,
rα({aβb}) is essential for all b∈Sβ(a).
We know that the intersection of finitely many essential right ideal is non zero.
Since Sβ(a) is finite, Tb∈Sβ(a)rα({aβb})6= 0.Hence rα({aβb/b ∈Sβ(a)})6= 0,
which contradicts to the β−insulator Sβ(a).Consequently, A= 0.
Theorem 5.8. If Mis a right (left) strongly prime Γ−ring, then the left (right)
operator ring L(R)is right (left) strongly prime ring.
Proof. Suppose that Mis right strongly prime Γ−ring. To prove Lis right
strongly prime ring, it is enough to prove that every non zero element in Lhas
a right insulator. Let Pi[xi, αi]6= 0 ∈L. Then there exists x∈Msuch that
Pi[xi, αi]x6= 0,that is Pixiαix6= 0.Since Mis right strongly prime, for every
β∈Γ, there exist an β−insulator for Pixiαix, say it Sβ={a1, a2, . . . , an}.

Strongly Prime Gamma Rings and Morita Equivalence of Rings 1145
Then
rαÃ(X
i
xiαixβc/c ∈Sβ)!={0},∀α, β ∈Γ.
Hence for any m∈M,
ÃX
i
xiαix!βakαm = 0,∀α, β ∈Γ, ak∈Sβ⇒m= 0.(1)
Now fix α, β ∈Γ,consider the collection
Sβ0={[xβa1, α],[xβa2, α], . . . , [xβan, α]}.
We shall prove that Sβ0is an insulator for Pi[xi, αi].It is enough to prove
that
Ann Ã(X
i
[xi, αi]c0/c0∈Sβ0)!={0}.
Let Pj[yj, βj]∈Ann ({Pi[xi, αi]c0/c0∈Sβ0}).Then Pi[xi, αi] [xβak, α]
Pj[yj, βj] = 0 for any k. We claim that Pj[yj, βj] = 0.Now
X
i
[xi, αi] [xβak, α]X
j
[yj, βj] = 0,∀k
implies that

X
i
[xi, αi] [xβak, α]X
j
[yj, βj]
m= 0,∀m∈M.
Therefore Pi[xi, αi] [xβak, α]Pj[yj, βj] (m) = 0,i.e., Pi[xiαixβak, α]
Pjyjβjm= 0,i.e., PixiαixβakαPjyjβjm= 0.By (1), Pjyjβjm= 0,i.e.,
Pj[yj, βj] (m) = 0,∀m∈M. Hence Pj[yj, βj] = 0.Since Pi[xi, αi]6= 0 is
arbitrary, every non zero element in Lhas a right β-insulator. Similarly if M
is left strongly prime, then every non-zero element of Rhas a left β-insulator.
Thus Lis right strongly prime and Ris a left strongly prime ring.
Theorem 5.9. If a Γ-ring Mis weakly semiprime then Mis strongly prime if
and only if its left operator ring Lis right strongly prime and its right operator
ring Ris left strongly prime.
Proof. Suppose that Lis a right strongly prime Γ-ring. In order to prove that
Mis a strongly prime Γ-ring, we shall prove that for every β∈Γ, every non
zero element in Mhas a right β−insulator. Let x6= 0 ∈M , β ∈Γ.Since Mis
a left weakly semiprime Γ−ring,[x, β]6= 0.Since Lis right strongly prime, there
exists a right insulator
S([x, β]) = 


n
X
j=1
[yjk, βjk] /k= 1,2, . . . , m 



1146 C. Selvaraj et al.
for [x, β].Then Ann ({[x, β]c/c∈S([x, Γ])}) = {0}.Therefore for any
P`[z`, δ`]∈L,
[x, β]
n
X
j=1
[yjk, βjk]X
`
[z`, δ`] = {0},for all k= 1,2, . . . , m
implies that
X
`
[z`, δ`] = 0.(2)
Consider S0
β={yjkβjkx/j= 1,2, . . . , n, k = 1,2, . . . , m }.We now claim that
S0
βis a right β-insulator for x. It is enough to prove that for each α∈Γ,
rα³nxβc/c ∈S0
βo´={0}.Let y∈rα³nxβc/c ∈S0
βo´,∀α∈Γ; then
(xβyjkβjkx)αy = 0,∀α∈Γ and k= 1,2,3, . . . , m.
Therefore
[xβyjkβjkxαy, Γ] = 0,∀α∈Γ and k= 1,2,3, . . . , m.
Hence [xβyjk, βjk] [xαy , Γ] = 0,∀α∈Γ and k= 1,2,3, . .. , m, that is
[x, β][yjk, βjk] [xαy, Γ] = 0,∀α∈Γ and k= 1,2,3, . . . , m, so that
[x, β]Pn
j=1 [yjk, βjk] [xαy, Γ] = 0,∀α∈Γ and k= 1,2,3, . . . , m.
From (2), we have [xαy, Γ] = 0,∀α∈Γ,so that xαy = 0,∀α∈Γ.Since M
is faithful L−Rbimodule, we have y= 0.Since x6= 0 ∈Mis arbitrary, for
every β∈Γ,every non zero element in Mhas a right β−insulator. Hence M
is a right strongly prime Γ−ring. Similarly if Ris a left strongly prime Γ-ring
then Mis a left strongly prime Γ-ring.
Converse part follows from Theorem 5.8.
Proposition 5.10. If Mis strongly prime Γ−ring, then Mis weakly semiprime
Γ-ring.
Proof. Suppose that Mis a strongly prime Γ−ring. We shall prove that M
is a weakly semiprime Γ-ring. Let x6= 0 ∈M. It is enough to prove that
[x, Γ] 6= 0 and [Γ, x]6= 0.Suppose that [x, Γ] = 0.Since Mis a strongly
prime Γ-ring, for every β∈Γ there exists a finite subset Sβ(x) such that for
b∈M, {xβcαb/c ∈Sβ(x)}= 0,∀α∈Γ implies that b= 0.Now xβcαx =
[x, β]cαx = 0cαx = 0,∀β , α ∈Γ, c ∈Sβ(x).Hence x= 0, a contradiction. Thus
Mis a weakly semiprime Γ−ring.
Theorem 5.11. Let Mbe a strongly prime Γ-ring, Land Rbe its operator rings.
Then Land Rare Morita Equivalent.
Proof. It follows from Proposition 5.10 and Theorem 1 of [9] and Theorem 2.3
of [10].

Strongly Prime Gamma Rings and Morita Equivalence of Rings 1147
Acknowledgement. The authors wish to express their indebtedness and gratitude
to the referee for the helpful suggestions and valuable comments.
References
[1] F.W. Anderson, K.R. Fuller: Rings and Categories of Modules, GTM 13, Springer
Verlag, New York, 1974.
[2] W.E. Barnes: On the Γ-rings of Nubusawa, Pacific J. Math. 18(3), 411–422 (1966).
[3] W.E. Coppage, J. Luh: Radicals of Γ−rings, J. Math. Soc. Japan 23(1), 40–52
(1971).
[4] D. Handelman, J. Lawerence: Strongly prime rings, Trans. Amer. Math. Soc. 211,
209–223 (1975).
[5] A. Kaucikas, R. Wisbauer: On strongly prime rings and ideals, Comm. in Algebra
25(11), 5461–5473 (2000).
[6] S. Kyuno: On prime Γ−rings, Pacific J. Math. 75(1), 185–190 (1978).
[7] J. Luh:On the theory of simple Γ−rings, Michigon. Math. J. 16, 65–75 (1969)
[8] N. Nobusawa: On a generalization of the ring theory, Osaka J. Math. 1,81–89
(1964).
[9] N. Nobusawa: Γ−rings and Morita equivalence, Math. J. Okayama Univ. 26,
151–156 (1984).
[10] M. Parvathi, A. Ramakrishna Rao: Morita equivalence for a larger class rings,
Publications Mathematica Debrecen 35, 67–71 (1988).