Editing Dual number (section)

==Generalizations==
This construction can be carried out more generally: for a [[commutative ring]] {{mvar|R}} one can define the dual numbers over {{mvar|R}} as the [[quotient ring|quotient]] of the [[polynomial ring]] {{math|''R''[''X'']}} by the [[ideal (ring theory)|ideal]] {{math|(''X''<sup>2</sup>)}}: the image of {{mvar|X}} then has square equal to zero and corresponds to the element {{mvar|ε}} from above.

=== Arbitrary module of elements of zero square ===
There is a more general construction of the dual numbers. Given a [[commutative ring]] <math>R</math> and a module <math>M</math>, there is a ring <math>R[M]</math> called the ring of dual numbers which has the following structures:

It is the <math>R</math>-module <math>R \oplus M</math> with the multiplication defined by <math>(r, i) \cdot \left(r', i'\right) = \left(rr', ri' + r'i\right)</math> for <math>r, r' \in R</math> and <math>i, i' \in I.</math>

The algebra of dual numbers is the special case where <math>M = R</math> and <math>\varepsilon = (0, 1).</math>