RESEARCH ARTICLE

An elementary description of $K_1(R)$ without elementary matrices

T. Hüttemann^a, Z. Zhang<sup>b

a</sup> Queen's University Belfast, School of Mathematics and Physics, Mathematical Sciences Research Centre, Belfast BT7 1NN, UK
^b School of Mathematics, Beijing Institute of Technology, 5 South Zhongguancun Street, Haidian District, 100081 Beijing, P. R. China

Abstract: 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, invertible matrix, elementary matrix.

MSC: Primary 19B99; Secondary 16E20

Received: 18.03.2020

Language: English

DOI: 10.12958/adm1568

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