Definition:Dorroh Extension

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)