This paper develops a structure
theory for any directly finite, right ℵ0-continuous, regular ring R, generalizing the
corresponding theory for a right and left ℵ0-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 ℵ0-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.
