Author(s): Xu Zhongming
Pages: 21-26
Year: 1983 Issue:  3
Journal: Journal of Mathematical Research and Exposition
Keyword:  Γ-环; 稠密性; 忠实既约模; 右算子环; 除环; 右理想; 半单; 量空间; 全阵环; 亚直和;
Abstract: Let be an Γ-ring and M be an irreducible -moduie, for α∈, α∈Γ, we define T[a,a]: M→M by m T[a,a] = maa for all m∈#M. Let End(M) be the ring of all endomorphisms of the additive group of M. We define as usual End (M)={End(M) In this paper the following results are obtained.Theorem 1 If M be an irreducible -module, then its ring of endomorphisms D=End (M) is a division ring.Definition 1 A Γ-ring is said to act densely on M (or is said to be dense on M), if for every n and x1,…, xn in M which are linearly independent over D and any n elements y1,…,yn in M, there are elements α1,…,αk in　Γand α1,…,αk insuch that Theorem 2 Let be a primitive Γ-ring and let M be a faithful irreducible module. If D =End (M), then is a dense Γ-ring of linear transformations on M over D.Theorem 3 A( )= x = (0)} of the Γ-ring is (0), then is primitive if and only if it is dense.Theorem 4 Let R be the right operator ring of a primitive Γ-ring .Then for some division ring D, either R is isomorphic to Dn, the ring of all n×n matrices over D, or, for any integer k there exists a subring Rk of R which maps homomor-phically onto Dk.Theorem 5 Let R be the right operator ring of a simple right Artinian Then following conditions are equivalent mutually:(1) is a primitive Γ-ring, (2) is a simple Γ-ring, (3) is a prime Γ-ring,(4) A() = (0) and R Dn for some n, where De is the ring of all n×n matrices over some divison ring D.Theorem 6 If R is the right operator ring of a Γ-ring , tnen Γ-ring is a simple and right Artinian if and only if R is a simple, right Artinian and Au()