Unimodular Regularity, Stable Range, and Exchange Rings
Titel:
Unimodular Regularity, Stable Range, and Exchange Rings
Auteur:
Wei, Jiaqun
Verschenen in:
Communications in algebra
Paginering:
Jaargang 36 (2008) nr. 7 pagina's 2456-2465
Jaar:
2008-07
Inhoud:
Let R be an exchange ring. In this article, we show that the following conditions are equivalent: (1) R has stable range not more than n; (2) whenever x ∈ Rn is regular, there exists some unimodular regular w ∈ nR such that x = xwx; (3) whenever x ∈ Rn is regular, there exist some idempotent e ∈ R and some unimodular regular w ∈ Rn such that x = ew; (4) whenever x ∈ Rn is regular, there exist some idempotent e ∈ Mn(R) and some unimodular regular w ∈ Rn such that x = we; (5) whenever a(nR) + bR = dR with a ∈ Rn and b,d ∈ R, there exist some z ∈ Rn and some unimodular regular w ∈ Rn such that a + bz = dw; (6) whenever x = xyx with x ∈ Rn and y ∈ nR, there exist some u ∈ Rn and v ∈ nR such that vxyu = yx and uv = 1. These, by replacing unimodularity with unimodular regularity, generalize the corresponding results of Canfell (1995, Theorem 2.9), Chen (Chen 2000, Theorem 4.2 and Proposition 4.6, Chen 2001, Theorem 10), and Wu and Xu (1997, Theorem 9), etc.
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