This construction can be carried out more generally: for a commutative ring *R* one can define the dual numbers over *R* as the quotient *R*[*X*]/(*X*^{2}): the image of *X* then has square equal to zero. This ring and its generalisations play an important part in the algebraic theory of derivations and differential forms.

The dual numbers over any field form a commutative local ring; the maximal ideal consists of classes of the form *a* + *bX* where *a* ≠ 0.

The word "dual" in mathematics is used in several other meanings as well, see for instance dual space and dual polyhedron.