Module VIII·Article III·~1 min read
Commutative Algebra and Algebraic Geometry
Tensor Algebra and Sylow Theorems (Advanced)
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Commutative Algebra
Noetherian Rings
A ring $R$ is Noetherian (in terms of the ascending chain condition on ideals) if every ascending chain of ideals $I_1 \subseteq I_2 \subseteq \ldots$ stabilizes.
Emmy Noether in the 1920s established that this condition is the correct generalization of finiteness for rings.
Examples: fields, $\mathbb{Z}$, $K[x_1, \ldots, x_n]$ (Hilbert's basis theorem).
Hilbert's Basis Theorem: If $R$ is Noetherian, then $R[x]$ is Noetherian. Consequently, $K[x_1, \ldots, x_n]$ is Noetherian — every ideal is finitely generated.
Spectrum of a Ring
$\mathrm{Spec}(R) = {\text{prime ideals of } R}$. This is a topological space (Zariski topology): closed sets are $V(I) = {p: I \subseteq p}$.
$\mathrm{Spec}(\mathbb{Z}) = {(0)} \cup {(p): p \text{ is prime}}$ — a “geometric” object from number theory.
$\mathrm{Spec}(K[x]) = {(0)} \cup {(x-a): a \in K}$ — points of the line plus the “generic point”.
Hilbert’s Nullstellensatz
If $K$ is an algebraically closed field and $I \subseteq K[x_1, \ldots, x_n]$ is an ideal, then $V(I) = \varnothing \iff 1 \in I$.
More generally: the radical of an ideal $I$ equals the intersection of all maximal ideals containing $I$.
This connects algebra (ideals) to geometry (algebraic varieties $V(I)$) — the foundation of algebraic geometry.
Localization
Localization $R[S^{-1}]$ is the ring of “fractions” with denominators from the multiplicative set $S$. This allows one to “look” at the ring near a given ideal.
Localization of $\mathbb{Z}$ at a prime $p$: $\mathbb{Z}_{(p)} = \left{a / b: p \nmid b\right}$ — the ring of $p$-integers.
In geometry: localization of the ring of functions near a point.
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