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# Integral domain

An integral domain is a commutative domain.

More explicitly a ring, $R$, is an integral domain if:

• it is commutative,
• $0\neq 1$ (where $0$ and $1$ are the additive and multiplicative identities, respectively)
• and it contains no zero divisors (i.e. there are no nonzero $x,y\in R$ such that $xy = 0$).

## Examples

Some common examples of integral domains are:

• The ring $\mathbb{Z}$ of integers.
• Any field.
• The p-adic integers, $\mathbb{Z}_p$.
• For any integral domain, $R$, the polynomial ring $R[x]$ is also an integral domain.
• Any finite integral domain is a field.