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In page Division ring:

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Much of linear algebra may be formulated, and remains correct, for modules over a division ring D instead of vector spaces over a field. Doing so, one must specify whether one is considering right or left modules, and some care is needed in properly distinguishing left and right in formulas. In particular, every module has a basis, and Gaussian elimination can be used. So, everything that can be defined with these tools works on division algebras. Matrices and their products are defined similarly.[citation needed] However, a matrix that is left invertible need not to be right invertible, and if it is, its right inverse can differ from its left inverse. (See Generalized inverse § One-sided inverse.)