Vol. 100, No. 1, 1982

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Directly finite aleph-nought-continuous regular rings

Kenneth R. Goodearl

Vol. 100 (1982), No. 1, 105–122

This paper develops a structure theory for any directly finite, right 0-continuous, regular ring R, generalizing the corresponding theory for a right and left 0-continuous regular ring. The key result is that R must be unit-regular, which also provides a proof, much simpler than previous proofs, that any right and left 0-continuous regular ring is unit-regular. It is proved, for example, that R modulo any maximal two-sided ideal is a right self-injective ring. The Grothendieck group K0(R) is shown to be a monotone σ-complete interpolation group, which leads to an explicit representation of K0(R) as a group of affine continuous real-valued functions on the space of pseudo-rank functions on R. It follows, for example, that the isomorphism classes of finitely generated projective right R-modules are determined modulo the maximal two-sided ideals of R. For another example, if every simple artinian homomorphic image of R is a t × t matrix ring (for a fixed t N), then R is a t × t matrix ring.

Mathematical Subject Classification
Primary: 16A30, 16A30
Secondary: 16A54
Received: 11 January 1981
Published: 1 May 1982
Kenneth R. Goodearl
University of California, Santa Barbara
United States