Elementary matricies (Ex)
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Let $R$ be an associative ring with unit and recall that an ''elementary matrix'' $E$ over $R$ is a square matrix of the form | Let $R$ be an associative ring with unit and recall that an ''elementary matrix'' $E$ over $R$ is a square matrix of the form | ||
$$ E = \text{Id}_n + a e_{ij} $$ | $$ E = \text{Id}_n + a e_{ij} $$ | ||
− | where $\text{Id}_n$ is the $n \times n$ identity matrix, $a \in R$ and $e_{ij}$ is the matrix with zeros in all places except $(i, j)$ where $i \neq j$. Clearly each elementary matrix | + | where $\text{Id}_n$ is the $n \times n$ identity matrix, $a \in R$ and $e_{ij}$ is the matrix with zeros in all places except $(i, j)$ where it is $1$ and we have $i \neq j$. Clearly each elementary matrix is invertible and so defines an element $E \in GL(R)$ where |
$$GL(R) = GL_{\infty}(R) : = \text{lim}_{n \to \infty}GL_n(R)$$ | $$GL(R) = GL_{\infty}(R) : = \text{lim}_{n \to \infty}GL_n(R)$$ | ||
is the limit of the invertible matricies. | is the limit of the invertible matricies. |
Revision as of 19:00, 23 March 2012
Let be an associative ring with unit and recall that an elementary matrix over is a square matrix of the form
where is the identity matrix, and is the matrix with zeros in all places except where it is and we have . Clearly each elementary matrix is invertible and so defines an element where
is the limit of the invertible matricies.
Exercise 0.1. Prove that , where is the subgroup generated by all elements in which are represented by elementary matrices.
Hint 0.2. For and write the matrix
as a product of elementary matrices
where is an matrix.
Recall that is defined to be the abelian group
Exercise 0.3. Prove that .