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Row versus column operations. (English) Zbl 0860.19002

Let \(R\) be a commutative ring with 1, and let \(L_n(R,R)\) denote the set of all \(n\times n\) matrices \(A\) for which any sequence of elementary row operations can be replaced by a sequence of elementary column operations having the same effect; i.e., \(L_n (R,R)= \{A\in M_n(R): E_n(R)A \subseteq A E_n(R)\}\) where \(E_n(R)\) is the subgroup of \(GL_n (R)\) generated by matrices of the form \(E_{ij} (r)\), \(i\neq j\) (here \(E_{ij} (r)\) is the matrix with 1’s on the diagonal, \(r\) in the \((i,j)\)-position and zeroes everywhere else). Clearly if \(E_n(R)\) is normal in \(GL_n(R)\) then \(L_n(R,R) \supseteq GL_n(R)\). This is true if \(n\geq 3\) or if \(n=2\) and certain ‘stock conditions’ (listed in the paper under review) hold. Under these conditions it is easy to see that \(L_n(R,R) \supseteq R\cdot GL_n(R)\). If \(R\) is a field then equality holds here.
Suppose now that \(R\) is an integral domain and \(n\geq 3\) or the stock conditions hold. The authors prove the following facts:
If \(A\in L_n(R,R)\) and if \(\text{det} (A)=0\), then \(A=0\).
If \(A\neq 0\) then \(A\in L_n (R,R)\) if and only if \(\langle A \rangle\cdot\langle \text{adj} (A) \rangle =\langle \text{det} (A)\rangle\) where \(\langle A\rangle\) denotes the ideal of \(R\) generated by the entries of the matrix \(A\).
If \(A\neq 0\) and if \(A\in L_n (R,R)\) then the ideal \(\langle A\rangle\) is invertible and represents a class \([A]\) in \(\text{Pic} (R)\) of order dividing \(n\). Furthermore \([A]=0\) if and only if \(A\in R \cdot GL_n(R)\).
The authors give examples of matrices in \(L_n(R,R)-R \cdot GL_n(R)\). Some of their results apply to more general commutative rings.

MSC:

19B14 Stability for linear groups
20G15 Linear algebraic groups over arbitrary fields
15A30 Algebraic systems of matrices
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