Zero-product Property - Examples

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

Other articles related to "examples":

Destroyer - Post-war
... Examples include the British Daring class, US Forrest Sherman-class, and the Soviet Kotlin-class destroyers ... Examples include the US FRAM I programme and the British Type 15 frigates converted from fleet destroyers ... Examples include the Soviet Kashin-class, the British County class, and the American Charles F ...
English Orthography - Spelling Irregularities - "Ough" Words
... though /oʊ/ as in toe (other examples dough) tough /ʌf/ as in cuff (other examples rough, enough) cough /ɒf/ as in off (other examples Gough (name, some ...
Proverbs in Advertising
... Tatira has given a number of examples of proverbs used in advertising in Zimbabwe ... However, unlike the examples given above in English, all of which are anti-proverbs, Tatira's examples are standard proverbs ... to make a potential customer smile, in one of the Zimbabwean examples "both the content of the proverb and the fact that it is phrased as a proverb secure the idea of a ...

Famous quotes containing the word examples:

    Histories are more full of examples of the fidelity of dogs than of friends.
    Alexander Pope (1688–1744)

    There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.
    Bernard Mandeville (1670–1733)

    No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.
    André Breton (1896–1966)