A skewfield (also spelled skew-field) or a division ring is a unital ring where each nonzero element has an inverse (but zero does not). A commutative skewfield is called a field, but sometimes in specialized works on skewfields one often says simply field for a skewfield.
The linear algebra is often understood in the generality of division rings, namely the usual notions of basis, dimension, linear map, matrix of a linear map with respect to two bases and so on, and even Gauss elimination procedure, hold without changes for left or right vector spaces over a division ring.
The most famous noncommutative example is the skewfield of quaternions. The Frobenius theorem? states that apart from the fields of real and complex numbers and quaternions, there are no associative finite-dimensional division algebras over the real numbers; and even if one includes nonassociative finite-dimensional division algebras one obtains only one more example (the octonions).