Since R is an integral domain, the sum must be zero.
If however is not an integral domain, then the conclusion need not hold.
On the other hand, not every reduced ring is an integral domain.
A regular ring need not be an integral domain.
The number ω is a unit in that integral domain.
Every integral domain can carry at least the trivial absolute value.
Let R be an algebra over a field k that is an integral domain.
For a similar reason any integral domain that is not a field is unstable.
That is, let a, b, and c belong to an integral domain.
The characteristic of every integral domain is either zero or a prime number.
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