# Definition:Ring Zero

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## 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**.

When the ring in question is also a field, the **ring zero** is called the **field 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): Chapter $\text {I}$: Elementary Definitions: $\S 1$. Rings and Fields - 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