# Definition:Dorroh Extension

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## Definition

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 $\struct {R \times \Z, +, \cdot}$.

## Also known as

The **Dorroh extension** is also known as the **unitization**.

## Also see

- Dorroh Extension is Ring with Unity
- Definition:Unitization Functor
- Ring can be Embedded in Dorroh Extension
- Every Ring can be Embedded in Ring with Unity

## Source of Name

This entry was named for Joe Lee Dorroh.

## Sources

- 1932: J.L. Dorroh:
*Concerning adjunctions to algebras*(*Bull. Amer. Math. Soc.***Vol. 38**: pp. 85 – 88)