\begin{definition}[Definition:Dorroh Extension]
Let $R$ be a ring.
We define two operations on the cartesian product $R \times \Z$ as:
:$\tuple {r, n} + \tuple {s ,m} = \tuple {r + s, n + m}$
:$\tuple {r, n} \cdot \tuple {s, m} = \tuple {r s + n s + m r, n m}$
The '''Dorroh extension''' of $R$ is the ring $\struct {R \times \Z, +, \cdot}$.
\end{definition}