Properties
- Let R be an integral domain. Then there is an integral domain S such that R ⊂ S and S has an element which is transcendental over R.
- The cancellation property holds in integral domains. That is, let a, b, and c belong to an integral domain. If a ≠ 0 and ab = ac then b = c. Another way to state this is that the function x ↦ ax is injective for any non-zero a in the domain.
- An integral domain is equal to the intersection of its localizations at maximal ideals.
- An inductive limit of integral domains is an integral domain.
Read more about this topic: Integral Domain
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)