By Alexander Schmitt

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**Extra resources for Algebra I: Commutative Algebra [Lecture notes]**

**Example text**

1) But then, deg(g) < deg( fk+1 ). , fk . , fk . , fk , and this is a contradiction. , ak )k≥1 is a non-stationary ascending chain in R. This contradicts the assumption that R is noetherian. 7 Corollary. , a field. , xn] is noetherian, n ≥ 1. 8 Remark (An application to algebraic sets). , xn] a possibly infinite subset. , fm } . 1. Chain Conditions In particular, every algebraic set may be defined by finitely many equations. 1, v), V(F) = V F . , xn] is a noetherian ring, so that F is finitely generated.

An, u) = R⋆ . , an). , an) holds if and only if a1 = · · · = an = 0. Proof. 3, vi). , an) ⇐⇒ d ⊃ a1 ∪ · · · ∪ an . The fact that d is an ideal implies d ⊃ a1 ∪ · · · ∪ an ⇐⇒ d ⊃ a1 + · · · + an . The rest of the asserted properties is straightforward to verify, and we leave the proofs to the reader. , an) ⊂ R⋆ . , an) = R⋆ . , an) : d ′ |d. , an. 6. 8 Properties. , an ∈ R be ring elements. , an) and d ∼ d ′. , an ). , a greatest common divisor is determined up to units. , an). , a′n ∈ R with ai = d · a′i are coprime.

6. Factorial Rings property: For every field K and every injective homomorphism ϕ : R −→ K, there is a unique homomorphism Φ : Q(R) −→ K with Φ ◦ ι = ϕ. Φ ❴ ❴ ❴ ❴ ❴ ❴ ❴ / K. Q(R) The universal property expresses that Q(R) is the smallest field that contains R. In order to construct it, we obviously have to invert the elements of R \ {0}. 5). For (a, b), (c, d) ∈ R × (R \ {0}), we write (a, b) ∼ (c, d) :⇐⇒ a · d = b · c. 5 Proposition. The relation “∼” is an equivalence relation on R × (R \ {0}).

### Algebra I: Commutative Algebra [Lecture notes] by Alexander Schmitt

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