Rings and Fields¶
Job: commutative algebra toolkit for algebraic geometry and number theory — ideals, localization, Noetherian conditions, spectrum.
Core¶
Definition. A ring \(R\) here means associative, unital, not necessarily commutative unless stated; homomorphisms send \(1\mapsto 1\). A field is a commutative ring with \(R^\times=R\setminus\{0\}\).
| Notion | Statement |
|---|---|
| Ideal \(\mathfrak{a}\subseteq R\) | \(R\mathfrak{a}R\subseteq\mathfrak{a}\) (two-sided); for commutative: \(R\mathfrak{a}\subseteq\mathfrak{a}\) |
| Quotient | \(R/\mathfrak{a}\); prime \(\mathfrak{p}\) \(\Leftrightarrow\) \(R/\mathfrak{p}\) domain; maximal \(\mathfrak{m}\) \(\Leftrightarrow\) \(R/\mathfrak{m}\) field |
| Chinese remainder | \(\mathfrak{a}+\mathfrak{b}=R\) \(\Rightarrow\) \(R/(\mathfrak{a}\cap\mathfrak{b})\cong R/\mathfrak{a}\times R/\mathfrak{b}\) |
| Localization \(S^{-1}R\) | \(S\) multiplicative; elements \(r/s\); universal inversion of \(S\) |
| \(R_{\mathfrak{p}}\) | \(S=R\setminus\mathfrak{p}\); local ring with maximal ideal \(\mathfrak{p}R_{\mathfrak{p}}\) |
| Noetherian | ACC on ideals \(\Leftrightarrow\) every ideal f.g. |
| Hilbert basis | \(R\) Noetherian \(\Rightarrow\) \(R[x_1,\ldots,x_n]\) Noetherian |
| PID | every ideal principal; PID \(\Rightarrow\) UFD; Euclidean \(\Rightarrow\) PID |
| Field extension \(L/K\) | \(L\) field containing \(K\); finite \(\Leftrightarrow\) \([L:K]<\infty\) as \(K\)-vector spaces |
Localization (exactness)¶
For commutative \(R\), localization is exact: if \(0\to M'\to M\to M''\to 0\) is exact in \(R\)-Mod, so is \(0\to S^{-1}M'\to S^{-1}M\to S^{-1}M''\to 0\). In particular, \(M=0\) iff \(M_{\mathfrak{m}}=0\) for all maximal \(\mathfrak{m}\) (local-to-global for vanishing).
Spec intuition¶
\(\operatorname{Spec} R=\{\text{prime ideals of }R\}\) with Zariski topology: closed sets \(V(\mathfrak{a})=\{\mathfrak{p}:\mathfrak{a}\subseteq\mathfrak{p}\}\). Basic opens \(D(f)=\{\mathfrak{p}:f\notin\mathfrak{p}\}\).
| Algebra | Geometry |
|---|---|
| \(R\to S\) ring map | \(\operatorname{Spec} S\to\operatorname{Spec} R\) continuous |
| \(\mathfrak{p}\) prime | point; \(\overline{\{\mathfrak{p}\}}=V(\mathfrak{p})\) |
| nilradical \(\sqrt{(0)}\) | functions vanishing on all of \(\operatorname{Spec}\) |
| \(R\) domain | \(\operatorname{Spec}\) irreducible |
| localization \(R_f\) | functions on \(D(f)\) |
Fields and algebraic extensions¶
\(L/K\) algebraic if every \(\alpha\in L\) is a root of some nonzero \(f\in K[x]\). Finite \(\Rightarrow\) algebraic. Splitting field of \(f\) over \(K\) exists and is unique up to \(K\)-isomorphism. Separable: minimal polynomials have distinct roots (automatic in char \(0\)). Normal: every irr. \(f\in K[x]\) with a root in \(L\) splits completely in \(L\).
Theorem. Finite \(L/K\) is Galois (normal + separable) iff \(\|\operatorname{Aut}_K(L)\|=[L:K]\). Then write \(\operatorname{Gal}(L/K):=\operatorname{Aut}_K(L)\).
Standard examples¶
| Ring | Properties |
|---|---|
| \(\mathbb{Z}\) | Euclidean domain, Dedekind |
| \(k[x]\) | Euclidean if \(k\) field |
| \(\mathbb{Z}[x]\) | UFD, not PID (\((2,x)\)) |
| \(k[x,y]\) | UFD, not PID; \(\dim=2\) |
| \(\mathbb{Z}/n\mathbb{Z}\) | Artinian; product of primary via CRT |
| \(\mathcal{O}_K\) | Dedekind domain (number fields) |
See it¶
Nullstellensatz (alg. closed \(k\)). Maximal ideals of \(k[x_1,\ldots,x_n]\) are \((x_1-a_1,\ldots,x_n-a_n)\); radical ideals \(\leftrightarrow\) algebraic sets in \(\mathbb{A}^n\). This is the bridge \(R\leftrightarrow\operatorname{Spec} R\) that algebraic geometry packages as schemes (glue local models \(\operatorname{Spec} R\)).
Dedekind domains. Noetherian integrally closed domains of dim \(1\): every nonzero ideal factors uniquely into primes — the correct generalization of unique factorization when \(\mathcal{O}_K\) fails to be a PID (e.g. \(\mathbb{Z}[\sqrt{-5}]\)).
Check¶
Show \(\mathbb{Z}_{(p)}\) is a DVR: what is the unique nonzero prime ideal?
Localize \(\mathbb{Z}\) at \(S=\mathbb{Z}\setminus p\mathbb{Z}\). Maximal (and only nonzero prime) is \(p\mathbb{Z}_{(p)}\); every nonzero ideal is \((p^k)\). Uniformizer \(p\).
Counterexample: localization need not preserve being a domain? Wait — it does if \(0\notin S\). Give instead a map that is injective but not injective after \(\otimes_R R/\mathfrak{a}\).
\(R=\mathbb{Z}\), \(\times 2:\mathbb{Z}\to\mathbb{Z}\) is injective; mod \(2\) it is zero. Localization at \(S=\{1,2,4,\ldots\}\) makes \(\times 2\) an iso on \(S^{-1}\mathbb{Z}=\mathbb{Z}[1/2]\). Point: different base changes have different exactness.
Pitfalls¶
- Prime vs maximal: \((0)\) prime in domain, not maximal if not a field. In \(\mathbb{Z}\), \((p)\) maximal; in \(k[x,y]\), \((x)\) prime not maximal.
- Noetherian \(\not\Rightarrow\) Artinian: \(\mathbb{Z}\) has ACC not DCC on ideals.
- UFD \(\not\Rightarrow\) PID: \(\mathbb{Z}[x]\), \(k[x,y]\).
- Integral closure is subtle: \(\mathbb{Z}[\sqrt{5}]\) not integrally closed in \(\mathbb{Q}(\sqrt{5})\); the ring of integers is \(\mathbb{Z}[\frac{1+\sqrt{5}}{2}]\).
- Noncommutative: simple Artinian rings are \(M_n(D)\) (Artin–Wedderburn); “ideals” mean two-sided for quotients to be rings.
Refs¶
- Atiyah–Macdonald, Introduction to Commutative Algebra
- Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry
- Lang, Algebra
Next¶
Modules · Galois theory · Group theory · Abstract algebra · Algebraic topology