Properties
Assuming f : A → B is a unital ring homomorphism, is an ideal in A, is an ideal in B, then:
- is prime in B is prime in A.
-
- It is false, in general, that being prime (or maximal) in A implies that is prime (or maximal) in B. Many classic examples of this stem from algebraic number theory. For example, embedding . In, the element 2 factors as where (one can show) neither of are units in B. So is not prime in B (and therefore not maximal, as well). Indeed, shows that, and therefore .
On the other hand, if f is surjective and then:
- and .
- is a prime ideal in A is a prime ideal in B.
- is a maximal ideal in A is a maximal ideal in B.
Read more about this topic: Extension And Contraction Of Ideals
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)