Dieudonné determinant

In linear algebra, the Dieudonné determinant is a generalization of the determinant of a matrix to matrices over division rings and local rings. It was introduced by Dieudonné (1943).

If K is a division ring, then the Dieudonné determinant is a group homomorphism from the group GL_n(K ) of invertible n-by-n matrices over K onto the abelianization K^×/ [K^×, K^×] of the multiplicative group K^× of K.

For example, the Dieudonné determinant for a 2-by-2 matrix is the residue class, in K^×/ [K^×, K^×], of

${\displaystyle \det \left({\begin{array}{*{20}c}a&b\\c&d\end{array}}\right)=\left\lbrace {\begin{array}{*{20}c}-cb&{\text{if }}a=0\\ad-aca^{-1}b&{\text{if }}a\neq 0.\end{array}}\right.}$