Geodesic
An elementary description of $K_1(R)$ without elementary matrices
Algebra and discrete mathematics, Tome 30 (2020) no. 1, pp. 79-82.

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Let $R$ be a ring with unit. Passing to the colimit with respect to the standard inclusions $\mathrm{GL}(n,R) \to \mathrm{GL}(n+1,R)$ (which add a unit vector as new last row and column) yields, by definition, the stable linear group $\mathrm{GL}(R)$; the same result is obtained, up to isomorphism, when using the “opposite” inclusions (which add a unit vector as new first row and column). In this note it is shown that passing to the colimit along both these families of inclusions simultaneously recovers the algebraic $K$-group $K_1(R) = \mathrm{GL}(R)/E(R)$ of $R$, giving an elementary description that does not involve elementary matrices explicitly.
Keywords: $K$-theory, elementary matrix.
Mots-clés : invertible matrix 
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T. Hüttemann; Z. Zhang. An elementary description of $K_1(R)$ without elementary matrices. Algebra and discrete mathematics, Tome 30 (2020) no. 1, pp. 79-82. http://geodesic.mathdoc.fr/item/ADM_2020_30_1_a6/

[1] Hyman Bass, Algebraic {$K$}-theory, W. A. Benjamin, Inc., New York–Amsterdam, 1968 | MR | Zbl