# Definition:Null Ring

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## Definition

A ring with one element is called **the null ring**.

That is, the **null ring** is $\struct {\set {0_R}, +, \circ}$, where ring addition and the ring product are defined as:

\(\ds 0_R + 0_R\) | \(=\) | \(\ds 0_R\) | ||||||||||||

\(\ds 0_R \circ 0_R\) | \(=\) | \(\ds 0_R\) |

## Also known as

Some authors refer to this as the **zero ring**, others as the **degenerate ring**.

Still others refer to it as **the trivial ring**, but this term has been defined differently elsewhere.

## Also see

- Null Ring is Trivial Ring in which it is seen that the
**null ring**is a trivial ring and therefore a commutative ring. - Uniqueness of Null Ring

- Results about
**the null ring**can be found**here**.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $21$. Rings and Integral Domains: Example $21.4$ - 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): $\S 3.1$: Direct sums - 2008: Paul Halmos and Steven Givant:
*Introduction to Boolean Algebras*... (previous) ... (next): $\S 1$