By Felsner S.
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Example text
1, R has stable range one. 6. Let n ≥ 2 be a positive integer. Then the following are equivalent: (1) If A ∈ Mn (R), then A can be written as A = W LU, W ∈ W, L ∈ L, U ∈ U and in L and U all the diagonal entries are equal to 1. (2) R is a right hermitian ring having stable range one. Proof. 5, R has stable range one. It will suffice to prove that R is a right hermitian ring. Let a, b ∈ R. Then there exist c1 , c2 , · · · , cn ∈ R such that 1 ∗ ∗ ··· ∗ c1 1 0 0 0 ··· 0 1 ∗ ··· ∗ ∗ 1 0 0 0 · · · 0 ∗ c2 ∗ ∗ 1 ..
Rings December 8, 2010 10:20 World Scientific Book - 9in x 6in Chapter 2 Unit 1-Stable Range An associative ring R is said to have unit 1-stable range if aR + bR = R implies there exists a u ∈ U (R) such that a + bu ∈ U (R). Many authors have studied this condition such as Chen ([85]), Goodearl and Menal ([224]), and Menal and Moncasi ([321]). 2]. A ring R is said to satisfy the Goodearl-Menal condition provided that for any x, y ∈ R there exists some u ∈ U (R) such that x − u, y − u−1 ∈ U (R).
Hence, ∗ ∗ ··· ∗ ∗ ∗ ··· 1 n×n n×n 00 0 0 .. Γ, . 1 0 ∗1 ∗ ∗ .. 1 . n×n ∗ 1 1 ∗ ··· ∗ ∗ ∗ ∗ 1 1 ··· ∗ U =. . Γ ·Γ Γ Γ . . . .. .. . .. . . ∗ ∗ ··· ∗ ∗ ∗ · · · 1 n×n 1 1 1 ∗ ··· ∗ ∗ ∗ 1 1 ··· ∗ ∗ ∗ . =. . . . . . .. .. . .. . 1, R has stable range one. 6. Let n ≥ 2 be a positive integer. Then the following are equivalent: (1) If A ∈ Mn (R), then A can be written as A = W LU, W ∈ W, L ∈ L, U ∈ U and in L and U all the diagonal entries are equal to 1.
3-Interval irreducible partially ordered sets by Felsner S.
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