- A ring in which the zero-product property holds is called a domain. A commutative domain with a multiplicative identity element is called an integral domain. Any field is an integral domain; in fact, any subring of a field is an integral domain (as long as it contains 1). Similarly, any subring of a skew field is a domain. Thus, the zero-product property holds for any subring of a skew field.
- If is a prime number, then the ring of integers modulo has the zero-product property (in fact, it is a field).
- The Gaussian integers are an integral domain because they are a subring of the complex numbers.
- In the strictly skew field of quaternions, the zero-product property holds. This ring is not an integral domain, because the multiplication is not commutative.
- The set of nonnegative integers is not a ring, but it does satisfy the zero-product property.
Read more about this topic: Zero-product Property
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