X. Principal Ideal Domains

Question 1

Definition
Ideal of ring.

Question 2

Lemma X.1.1
An ideal I = R iff.

Question 3

Definition
Sum of ideals.
Product of ideals.

Question 4

Lemma X.1.2
Properties of ideals.

Question 5

Definition
Set of multiples.

Question 6

Lemma X.1.3
a \in R, then Ra is…

Question 7

Definition
Principal ideal generated by…
Principal.

Question 8

Definition
Ideal generated by…

Question 9

Lemma X.2.1
equivalences to b \in Ra

Question 10

Lemma X.2.2
Ra = Rb iff.
Ra = R iff.

Question 11

Definition
Principle Ideal Domain

Question 12

Theorem X.4.1
If R is a Euclidean domain…

Question 13

Corollary X.4.4
For any field K, K[x] is…

Question 14

Theorem X.5.1
In a P.I.D., the primes of R are…

Question 15

Lemma X.5.3
In a P.I.D. unique factorisation…

Question 16

Definition
Increasing sequence of ideals.
Stationary sequence.
Noetherian ring.

Question 17

Lemma X.6.1
Union of increasing sequence of ideals…

Question 18

Lemma X.6.2
If R is a P.I.D., then R is…

Question 19

Lemma X.6.3
A sequence of elements in a P.I.D. s.t. they increasing divide each other…

Question 20

Lemma X.6.4
Every non-zero, non-unit element in a P.I.D…