# Integers form Commutative Ring

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## Theorem

The set of integers $\Z$ forms a commutative ring under addition and multiplication.

## Proof

We have that:

- From Integers under Addition form Abelian Group, the algebraic structure $\struct {\Z, +}$ is an abelian group.

- From Integers under Multiplication form Countably Infinite Commutative Monoid, the algebraic structure $\struct {\Z, \times}$ is a commutative monoid and therefore a commutative semigroup.

Thus all the ring axioms are fulfilled, and $\struct {\Z, +, \times}$ is a commutative ring.

By Integer Multiplication has Zero, the zero is $0$.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $21$. Rings and Integral Domains: Example $21.1$ - 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $1$: The definition of a ring: Definitions $1.1 \ \text{(c)}$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 55$. Special types of ring and ring elements: $(1)$

- 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 4$: Number systems $\text{I}$: The rational integers