Abstract
In this paper we prove two main results. The first one is the existence of a right self-injective regular ring of type III with center any given commutative self-injective regular ring. The second one is a characterization of the lattice of two-sided ideals for a large class of right self-injective regular rings of type III. In a previous paper, this lattice was described as the lattice of order ideals of a certain lattice of continuous functions. Now we prove the converse: for a large class of complete boolean spaces X, let C be the set of all continuous functions from X into {0} ∪ [ℵ0, γ], where γ is any infinite cardinal number, and let Δ be the lattice of all functions of C that are less than or equal to a given nonvanishing function of C. Then there exists a right self-injective regular ring R of type III such that the lattice of two-sided ideals of R is isomorphic to the lattice of order ideals of Δ.
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