Zero-product Property - Examples


  • 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.

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