# Definition:Ring Zero

## Contents

## Definition

Let $\struct {R, +, \circ}$ be a ring.

The identity for ring addition is called the **ring zero** (of $\struct {R, +, \circ}$).

It is denoted $0_R$ (or just $0$ if there is no danger of ambiguity).

## Also known as

When it is clear and unambiguous what is being discussed, the **ring zero** is often called just the **zero**.

In the context of number fields, the **ring zero** is sometimes seen referred to as the **additive identity**.

## Also see

- In Ring Product with Zero, it is shown that the ring zero is a zero element for the ring product, thereby justifying its name as the
**zero**of the ring.

## Sources

- 1964: Iain T. Adamson:
*Introduction to Field Theory*... (previous) ... (next): $\S 1.1$ - 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $1$: Integral Domains: $\S 3$. Definition of an Integral Domain - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 54$. The definition of a ring and its elementary consequences - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $0$ Zero - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**additive identity** - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $0$ Zero - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**additive identity**