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K 0 of Semiartinian Von Neumann Regular Rings. Direct Finiteness Versus Unit-Regularity

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Abstract

If R is a regular and semiartinian ring, it is proved that the following conditions are equivalent: (1) R is unit-regular, (2) every factor ring of R is directly finite, (3) the abelian group K O(R) is free and admits a basis which is in a canonical one to one correspondence with a set of representatives of simple right R-modules. For the class of semiartinian and unit-regular rings the canonical partial order of K O(R) is investigated. Starting from any partially ordered set I, a special dimension group G(I) is built and a large class of semiartinian and unit-regular rings is shown to have the corresponding K O(R) order isomorphic to G(P r i m R ), where P r i m R is the primitive spectrum of R. Conversely, if I is an artinian partially ordered set having a finite cofinal subset, it is proved that the dimension group G(I) is realizable as K O(R) for a suitable semiartinian and unit-regular ring R.

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Acknowledgments

The authors wish to express their gratitude to Ken Goodearl, who noticed a crucial mistake in a preliminary version of this work. His counterexample gave a clear indication about the right direction to follow in order to correct that mistake. The correction, however, resulted in a complete rewriting of the fourth section.

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Correspondence to Giuseppe Baccella.

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Presented by Kenneth Goodearl.

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Baccella, G., Spinosa, L. K 0 of Semiartinian Von Neumann Regular Rings. Direct Finiteness Versus Unit-Regularity. Algebr Represent Theor 20, 1189–1213 (2017). https://doi.org/10.1007/s10468-017-9682-3

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  • DOI: https://doi.org/10.1007/s10468-017-9682-3

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