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.
Read more about this topic: Zero-product Property
Famous quotes containing the word examples:
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (1533–1592)
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)