Page "Division ring" Paragraph 3

Much of linear algebra may be formulated, and remains correct, for ( left ) modules over division rings instead of vector spaces over fields.

Every module over a division ring has a basis ; linear maps between finite-dimensional modules over a division ring can be described by matrices, and the Gaussian elimination algorithm remains applicable.

Differences between linear algebra over fields and skew fields occur whenever the order of the factors in a product matters.

For example, the proof that the column rank of a matrix over a field equals its row rank yields for matrices over division rings only that the left column rank equals its right row rank: it does not make sense to speak about the rank of a matrix over a division ring.