Section 45 number 30 with Lemma 45.10 Lemma(Ascending Chain Condition for a PID) Let D be a PID. If N1 C N2 c... is anascending chain of ideals N;, then there exists a positive integer r such that N, = N, forall s > r. Equivalently, every strictly ascending chain of ideals (all inclusions proper) ina PID is of finite length. We express this by saying that the ascending chain condition(ACC) holds for ideals in a PID. poly30. Show that in a PID, every proper ideal is contained in a maximal ideal. [Hint: Use Lemma 45.10.]31 ctorinto irreducibles in Or vland prove that each of the factors is irreducible.There are several other concepts often considered that are similar in character to the ascending chain conditicideals in a Ting. The following three exercises concerm-some ef these concepts.

To prove - In a Principal ideal domain(PID), every proper ideal is contained in a maximal ideal.

Answered: 30. Show that in a PID, every proper…

30. Show that in a PID, every proper ideal is contained in a maximal ideal. [Hint: Use Lemma 45.10.| tho foctors is irreducible.

Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Section 45 number 30 with Lemma

45.10 Lemma
(Ascending Chain Condition for a PID) Let D be a PID. If N1 C N2 c... is an
ascending chain of ideals N;, then there exists a positive integer r such that N, = N, for
all s > r. Equivalently, every strictly ascending chain of ideals (all inclusions proper) in
a PID is of finite length. We express this by saying that the ascending chain condition
(ACC) holds for ideals in a PID.

poly
30. Show that in a PID, every proper ideal is contained in a maximal ideal. [Hint: Use Lemma 45.10.]
31 ctor
into irreducibles in Or vland prove that each of the factors is irreducible.
There are several other concepts often considered that are similar in character to the ascending chain conditic
ideals in a Ting. The following three exercises concerm-some ef these concepts.

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