The group of units of a simple ring. I (Q2539885)

The main results of this work concern the question of solvability of normal subgroups of the group of units of a simple ring with idempotent. Using theorems of Herstein on the Lie structure of a simple ring the following generalization of a theorem of Amitsur is proved. Theorem. Let \(R\) be a simple ring containing an idempotent \(e\ne 0,1\) and suppose the center \(Z\) of \(R\) has at least five elements. If \(W\) is a subspace of \(R\) invariant under all inner automorphisms by elements in \(N\), a normal subgroup of the units of \(R\), then \(N\subset Z\), or \(W\) is \(0\), \(Z\) or contains \([R,R]\). If \(W\) is a subalgebra of \(R\) then \(N\subset Z\) or \(W\) is \(0\), \(Z\) or \(R\). This theorem is then used to show that no normal subgroup of the units of \(R\) can be solvable unless it lies in \(Z\). The cases where \(Z\) has three or four elements are handled separately. The above theorem is further generalized to the case where \(N\) is a normal subgroup of a normal subgroup. In this case \(Z\) is assumed to have at least nine elements. This generalization yields the fact that normal subgroups of normal subgroups are not solvable unless they lie in \(Z\). The results for normal subgroups are extended to prime rings with idempotent and to semiprime rings with non-central idempotent which satisfy the property that any element centralized by all idempotents is in the center, as for example matrix rings over semiprime rings.