On Subdirectly Irreducible Completely Left Symmetric F-rings
Received:February 23, 1989  
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Fan Fusheng Dept. Math.
Northeast University of Technology
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      A left ideal L of a Г-ring M is called left symmetric if aabβb∈L implies aacβb∈L, where a,b, c∈ M, a ,β∈Γ. A Γ-ring M is called to be a Γ-division ring if it has the left unity, and if left oprator ring R is a division ring.In this paper, the following results are pvoved:Suppose that every left ideal of a Γ-ring M is left symmetiic.If A is subdirectly irreducible with more than one element and with no nonzero nilpotents then it is a Γ-division ring.If M is subdirectly irreducible and if D≠M then M/D is a Γ-division ring where D=(a∈M|aΓb=0,0≠b∈M).
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