Abstract Algebra | Ideals of quotients of PIDs

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We prove that every ideal of a quotient of a principal ideal domain is also principal. Notice that the new space may not be an integral domain, so it is sometimes called a principal ring.

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Small mistake (but I think important) in the second exercise: In a PID, every *nonzero* prime is a maximal ideal.

In an arbitrary integral domain, 0 is prime ideal but need not be maximal.

chaosjunks
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Warning: You can't just consider sequences like $I_1 \subset I_2 \dotsb$ when applying Zorn's Lemma. You have to consider an arbitrary totally ordered subset $(I_a)_{a \in A}$. Of course the argument goes through essentially unchanged once you make this small correction.

infphreak
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I thought you were going to make a course about PID controllers.

AhmedHan
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Was the Fourth Isomorphism Theroem covered in a previous video?

foreachepsilon
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This is a great video but too much of this requires some more foundation in Group Theory and other concepts of Algebra that I don't have. It would be great if there was some links in the description for primers on PID, and any of the other building blocks used in this video.

stevenwilson