A matrix is a rectangular array of quantities which, as an array, obeys certain rules when combined with other matrices by the operations of addition and multiplication, or when combined with scalar quantities by the operation of multiplication. These operations have meaning if and only if the matric quantities and scalar quantities are elements of a ring. A minor of a matrix is a certain function of a square sub-array of the matrix, and has a unique meaning if and only if the elements of the sub-array are commutative. The rank of a matrix is a function of all possible minors of the matrix, and thus a matrix will have a unique rank if and only if the matric quantities are elemtns of a commutative ring, say R.
Mills, Harlan D., "Constant Rank Matrices" (1950). The Harlan D. Mills Collection.