Abstract Algebra, Lec 25B: Ring Ideals, Factor (Quotient) Rings, Prime Ideals, Maximal Ideals

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(0:00) Lecture B will be theoretical with simple examples: make sure you work hard on the reading before Lecture 26.
(0:35) (Two-Sided) Ideals, proper ideals, "ideal test", and basic examples.
(8:43) Principal ideal generated by an element and examples (and similarities and differences with the idea of a cyclic subgroup generated by a group element).
(15:22) Factor (Quotient) Ring R/A where A is an ideal of R and an example Z/4Z.
(20:21) Prime ideals, maximal ideals, basic examples, and relationship to factor (quotient) rings R/A of a commutative ring with unity R.

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  1. Bill Kinney
  2. modern algebra
  3. ring theory
  4. group theory
  5. subring
  6. subring test
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I have a lot of questions from this lecture. Can you please clarify, Prof. Kinney?

At 13:27, I don't understand how the set of all polynomials can be generated by x, for instance, how do I get (x + 1) from x? It seems to me the generator must be 1 or some other unit.

At 26:03, isn't ⟨5⟩ ⊂ ⟨5, 7⟩ ⊂ ℤ ? How can ⟨5⟩ be maximal?

Also, I assumed quotient rings would be finite, but that can't be the case, because that would make both theorems around 27:59 contradictory, considering finite integral domains are always fields. Is that so?

lucasvella