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:
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—John Locke (1632–1704)
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—Ralph Waldo Emerson (1803–1882)