On the Centroid of the Prime Gamma Rings II

M. Ali ÖZTÜRK

Makale detayları

On the Centroid of the Prime Gamma Rings II

Yazarlar:

M. Ali ÖZTÜRK

Yayıncı:

TUBITAK

Dergi:

Turkish Journal of Mathematics 2001 Vol.25 Issue 3, pp.367-377

Belge Tipi:

Makale

Yayın Tarihi:

2/2001

Belge Dili:

Türkçe

Document ID (Org No.):

2000749054

Sıklık:

İki Aylık

eISSN:

1303-6149

pISSN:

1300-0098

Anahtar kelimeler:

Özet:

 

The aim of this paper is to study the properities of the  extended centroid of the prime G-rings.  Main results are the  following theorems: (1) Let M be a simple G-ring with  unity.  Suppose that for some a\neq 0 in M we have ag1  xg2 ab1 yb2a = ab1 yb2 ag1  xg2a for all x, y\in M and g1 ,g2  ,b1 ,b2 \in G.  Then M is isomorphic onto the  G-ring Dn,m, where Dn,m is the additive abelian group  of all rectangular matrices of type n\times m over a division ring  D and G is a nonzero subgroup of the additive abelian group  of all rectangular matrices of type m\times n over a division ring  D.  Furthermore M is the G-ring of all n\times n matrices  over the field CG.  (2) Let M be a prime G-ring  and CG the extended centroid of M.  If a and b are  non-zero elements in S=MG CG such that ag xb  b = bb xg a for all x \in M and b ,g \in  G, then a and b are CG-dependent.  (3) Let M  be prime GF-ring, Q quotient G-ring of M and  CG the extended centroid of M.  If q is non-zero element  in Q such that qg1 xg2qb1yb2q =  qb1yb2qg1xg2q for all x, y\in M,  g1 , g2, b1, b 2 \in G then  S  is a primitive G-ring with minimal right ( left ) ideal such  that eG S, where e is idempotent and CGG e is  the commuting ring of S on eG S.

 

The aim of this paper is to study the properities of the  extended centroid of the prime G-rings.  Main results are the  following theorems: (1) Let M be a simple G-ring with  unity.  Suppose that for some a\neq 0 in M we have ag1  xg2 ab1 yb2a = ab1 yb2 ag1  xg2a for all x, y\in M and g1 ,g2  ,b1 ,b2 \in G.  Then M is isomorphic onto the  G-ring Dn,m, where Dn,m is the additive abelian group  of all rectangular matrices of type n\times m over a division ring  D and G is a nonzero subgroup of the additive abelian group  of all rectangular matrices of type m\times n over a division ring  D.  Furthermore M is the G-ring of all n\times n matrices  over the field CG.  (2) Let M be a prime G-ring  and CG the extended centroid of M.  If a and b are  non-zero elements in S=MG CG such that ag xb  b = bb xg a for all x \in M and b ,g \in  G, then a and b are CG-dependent.  (3) Let M  be prime GF-ring, Q quotient G-ring of M and  CG the extended centroid of M.  If q is non-zero element  in Q such that qg1 xg2qb1yb2q =  qb1yb2qg1xg2q for all x, y\in M,  g1 , g2, b1, b 2 \in G then  S  is a primitive G-ring with minimal right ( left ) ideal such  that eG S, where e is idempotent and CGG e is  the commuting ring of S on eG S.

Makale detayları

On the Centroid of the Prime Gamma Rings II

Yazarlar:

M. Ali ÖZTÜRK

Yayıncı:

TUBITAK

Dergi:

Turkish Journal of Mathematics 2001 Vol.25 Issue 3, pp.367-377

Belge Tipi:

Makale

Yayın Tarihi:

2/2001

Belge Dili:

Türkçe

Document ID (Org No.):

2000749054

Sıklık:

İki Aylık

eISSN:

1303-6149

pISSN:

1300-0098

G-division ring

G-field

extented centroid

central closure.

Sayı

Makale detayları

On the Centroid of the Prime Gamma Rings II

Yazarlar: M. Ali ÖZTÜRK

Yayıncı: TUBITAK

Dergi: Turkish Journal of Mathematics 2001 Vol.25 Issue 3, pp.367-377

Belge Tipi: Makale

Yayın Tarihi: 2/2001

Belge Dili: Türkçe

Document ID (Org No.): 2000749054

Sıklık: İki Aylık

eISSN: 1303-6149

pISSN: 1300-0098

G-division ring

G-field

extented centroid

central closure.

Sayı

Yazarlar:

M. Ali ÖZTÜRK

Yayıncı:

TUBITAK

Dergi:

Turkish Journal of Mathematics 2001 Vol.25 Issue 3, pp.367-377

Belge Tipi:

Makale

Yayın Tarihi:

2/2001

Belge Dili:

Türkçe

Document ID (Org No.):

2000749054

Sıklık:

İki Aylık

eISSN:

1303-6149

pISSN:

1300-0098