AI Summary & Key Takeaways
AI InsightIf 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.
📌 3 Key Takeaways
Abstract
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.
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.