Semiring of Dual Numbers

The semiring of dual numbers over a Commutative Semiring $R$ is the Quotient Semiring $R[x]/x^{2}$ of the Polynomial Semiring $R[x]$ by the Principal Semiring Ideal generated by the polynomial $x^{2}$.

Properties

• Addition of dual numbers $({a}+{b}\varepsilon)+({c}+{d}\varepsilon)$ is defined componentwise.

• Multiplication of dual numbers $({a}+{b}\varepsilon)({c}+{d}\varepsilon)$ is defined ${ac}+{(ad+bc)}\varepsilon$.

• A dual number ${a}+{b}\varepsilon$ can be represented as a Matrix

  $\par\begin{pmatrix}a&b\\ 0&a\\ \end{pmatrix}\par$

  The basis element ${0}+{1}\varepsilon$ is then the matrix

  $\par\begin{pmatrix}0&1\\ 0&0\\ \end{pmatrix}\par$

  which squares to itself.

• The semiring of dual numbers is a special case of a Square-Zero Extension.