01.05.01 · foundations / commutative-algebra

Commutative algebra — Noetherian rings, localisation, and primary decomposition

shipped3 tiersLean: none

Anchor (Master): Atiyah-Macdonald Ch. 1–7; Eisenbud 1995, Commutative Algebra with a View Toward Algebraic Geometry; Matsumura 1989, Commutative Ring Theory

Intuition Beginner

A commutative ring is a number system where you can add, subtract, and multiply, and where multiplication does not care about order. The integers are the guiding example. Commutative algebra studies the ideals of such rings — the sets that measure divisibility — and the central surprise is that, for the rings that matter, the ideal structure is finite.

A ring is Noetherian when its ideals cannot keep growing forever. In the integers, every ideal is just the multiples of one number, so there is no room for an endless ascending tower of ideals. This finiteness is what makes computation possible: you can always reduce to a finite list of generators.

Localisation is the algebraic version of "allow division by some chosen elements." You build the rational numbers from the integers by allowing division by every nonzero integer. Localisation lets you choose which elements to invert — for instance, invert everything except the multiples of $5$ , and you get a ring focused on the prime $5$ .

Primary decomposition factors an ideal the way you factor an integer. The integer $60$ breaks into prime-power pieces $4$ , $3$ , $5$ , and the ideal of all multiples of $60$ breaks into corresponding ideal-pieces. The spectrum turns the prime ideals of a ring into the points of a geometric space, so algebraic questions become geometric ones.

Visual Beginner

The spectrum of the integers has one point for each prime number, plus one extra "generic" point for the zero ideal. A shape in this space is the set of primes dividing a fixed number.

Each prime $(2), (3), (5), \dots$ is a closed point; the zero ideal $(0)$ sits behind all of them. The shape for the number $30$ is the set of points ${(2), (3), (5)}$ — the primes that divide $30$ . This is the Zariski topology: closed sets record divisibility.

Worked example Beginner

Take the ideal $(60)$ in the integers — all multiples of $60$ . The number $60$ factors as $4 \times 3 \times 5$ . The matching fact for ideals is that $(60)$ is the overlap of the ideals $(4)$ , $(3)$ , and $(5)$ .

Check it with the number $120$ . Since $120$ is a multiple of $4$ , of $3$ , and of $5$ , it lies in all three ideals, so it lies in their overlap — and $120$ is a multiple of $60$ .

Now check $30$ . It is a multiple of $3$ and of $5$ , but not of $4$ , so it is not in $(4)$ , and so it is not in the overlap — and indeed $30$ is not a multiple of $60$ .

The overlap $(4) \cap (3) \cap (5)$ equals $(60)$ . That overlap is a primary decomposition: an ideal written as the overlap of simpler, prime-power pieces. This is the integer-side picture of the theorem proved in general below.

Check your understanding Beginner

Formal definition Intermediate+

A commutative ring $R$ (with $1$ ) is Noetherian if it satisfies any of three equivalent conditions ^{[Atiyah-Macdonald Ch. 6]}:

the ascending chain condition (ACC) on ideals: every chain $I_{1} \subseteq I_{2} \subseteq \dots$ of ideals of $R$ stabilises, meaning $I_{n} = I_{n + 1} = \dots$ for large enough $n$ ;
every ideal of $R$ is finitely generated;
every nonempty collection of ideals of $R$ has a maximal element (with respect to inclusion).

The equivalence of (1)–(3) is proved in the Full proof set below. The prototype is $Z$ , where every ideal is $n Z$ , and the polynomial ring $k [x]$ over a field, where every ideal is generated by a single polynomial (the greatest common divisor of its elements).

Localisation

Let $S \subseteq R$ be a multiplicatively closed subset: $1 \in S$ and $s t \in S$ whenever $s, t \in S$ . The localisation $S^{- 1} R$ is the ring of fractions $r / s$ with $r \in R$ , $s \in S$ , subject to the relation $r / s \sim r^{'} / s^{'}$ iff $t (r s^{'} - r^{'} s) = 0$ for some $t \in S$ ^{[Atiyah-Macdonald Ch. 3]}. The map $ι : R \to S^{- 1} R$ , $r \mapsto r /1$ , is the canonical ring homomorphism.

When $S = R ∖ p$ for a prime ideal $p$ , the localisation is written $R_{p}$ . It is a local ring: it has a unique maximal ideal, namely $p R_{p}$ . Localising at $p$ is the algebraic act of "zooming in on the point $p$ " of the spectrum.

The spectrum and the Zariski topology

The (prime) spectrum of $R$ is the set $Spec R$ of prime ideals of $R$ . For each ideal $I$ , define the closed set

V (I) = {p \in Spec R : I \subseteq p} .

These sets are the closed sets of the Zariski topology on $Spec R$ ^{[Atiyah-Macdonald Exercises 1.15, 1.17]}. They satisfy $V (I) \cup V (J) = V (I J) = V (I \cap J)$ and $⋃_{α} V (I_{α}) = V (\sum_{α} I_{α})$ , which is exactly what is needed for the closed sets to define a topology. The points corresponding to maximal ideals are the closed points; the minimal primes are the generic points of the irreducible components.

Primary ideals and primary decomposition

An ideal $q ⊊ R$ is primary if $ab \in q$ and $a \in / q$ together imply $b^{n} \in q$ for some $n \geq 1$ . Equivalently, every zero-divisor in $R / q$ is nilpotent. The radical $q = {r : r^{n} \in q for some n}$ is then a prime ideal $p$ , and $q$ is called $p$ -primary. A primary decomposition of an ideal $I$ is an expression

I = q_{1} \cap q_{2} \cap \dots \cap q_{n}

with each $q_{i}$ primary ^{[Atiyah-Macdonald Ch. 4]}. When $R$ is Noetherian every ideal admits one (the Lasker–Noether theorem, stated in Advanced results).

Counterexamples to common slips

A Noetherian ring need not be a domain. The finite ring $Z /6 Z$ is Noetherian (it has only finitely many ideals) yet has zero-divisors.
Not every ring is Noetherian. The polynomial ring $k [x_{1}, x_{2}, \dots]$ in countably many variables has the strictly ascending chain $(x_{1}) ⊊ (x_{1}, x_{2}) ⊊ \dots$ that never stabilises.
Localisation can collapse ideals. Inverting $3$ in $Z$ sends the ideal $(3)$ to the whole localised ring, because the generator $3$ becomes a unit.

Key theorem with proof Intermediate+

Theorem (Hilbert's basis theorem). If $R$ is a Noetherian ring, then the polynomial ring $R [x]$ is Noetherian. By induction, $R [x_{1}, \dots, x_{n}]$ is Noetherian, and so every finitely generated algebra over a Noetherian ring is Noetherian ^{[Hilbert 1890]}.

Proof. Let $I ◃ R [x]$ be an ideal. For each $n \geq 0$ , let $L_{n} (I)$ be the set of leading coefficients of polynomials of degree exactly $n$ belonging to $I$ , together with $0$ . Each $L_{n} (I)$ is an ideal of $R$ , and $L_{n} (I) \subseteq L_{n + 1} (I)$ : a polynomial $a x^{n + 1} + \dots$ of degree $n + 1$ in $I$ gives, on multiplying by $x^{- 1}$ ... — to avoid division, note that multiplying a degree- $n$ polynomial $a x^{n} + \dots \in I$ by $x$ produces a degree- $(n + 1)$ polynomial with the same leading coefficient $a$ in $I$ , so the containment holds. Because $R$ is Noetherian, the ascending chain $L_{0} \subseteq L_{1} \subseteq \dots$ stabilises at some $L_{d}$ , and each $L_{n}$ is finitely generated.

For each of the finitely many generators of the ideals $L_{0}, \dots, L_{d}$ , choose a witnessing polynomial in $I$ with that leading coefficient. Label these polynomials $f_{1}, \dots, f_{m}$ , with degrees $d_{i}$ and leading coefficients $a_{i}$ , and let $J = (f_{1}, \dots, f_{m}) \subseteq I$ . We claim $I = J + (I \cap M)$ , where $M \subset R [x]$ is the $R$ -submodule of polynomials of degree less than $d$ .

Given $g \in I$ of degree $e \geq d$ with leading coefficient $c$ , the chain has stabilised, so $c \in L_{e} = L_{d}$ . Write $c = \sum_{i} r_{i} a_{i}$ with $r_{i} \in R$ . Then

g - i \sum r_{i} x^{e - d_{i}} f_{i} \in I

has degree strictly less than $e$ , because the leading term $\sum_{i} r_{i} a_{i} x^{e}$ cancels $c x^{e}$ . Repeating, every $g \in I$ is congruent modulo $J$ to an element of degree less than $d$ , that is, to an element of $I \cap M$ .

Now $M ≅ R^{d}$ is a finitely generated $R$ -module. Since $R$ is Noetherian, every submodule of a finitely generated $R$ -module is finitely generated, so $I \cap M$ is generated over $R$ by finitely many polynomials $h_{1}, \dots, h_{ℓ}$ . Then $I = (f_{1}, \dots, f_{m}, h_{1}, \dots, h_{ℓ})$ is finitely generated. By condition (2) above, $R [x]$ is Noetherian. $□$

Bridge. This theorem builds toward 04.02.01, where Noetherian-ness of the coordinate ring becomes the "finite-type" hypothesis that makes a scheme Noetherian, and the ascending chain condition appears again in 21.02.07 as the defining property of the rings of integers of number fields (Dedekind domains are Noetherian), the foundational reason being that finite generation is preserved by every standard ring operation — quotients, localisations, and polynomial extensions. This is exactly the pattern that makes every finitely generated $k$ -algebra Noetherian, and the bridge is that geometric finiteness (a Noetherian scheme) reduces cleanly to algebraic finiteness (a Noetherian ring).

Exercises Intermediate+

Exercise 5 (medium, symbolic).

Prove that the radical $I = {r \in R : r^{n} \in I for some n \geq 1}$ of an ideal $I$ is itself an ideal, and that $I$ is the intersection of the prime ideals containing $I$ .

Hint

For closure under addition, use the binomial expansion of $(r + s)^{m + n}$ . For the prime-intersection claim, separate the two directions.

Answer

If $r^{m}, s^{n} \in I$ then $(r + s)^{m + n}$ expands to terms each containing a factor $r^{i} s^{j}$ with $i \geq m$ or $j \geq n$ , so each term lies in $I$ ; hence $r + s \in I$ . Closure under multiplication by $R$ is immediate. So $I$ is an ideal. If $p \supseteq I$ is prime then $r^{n} \in I \subseteq p$ forces $r \in p$ , giving $I \subseteq ⋂_{p \supseteq I} p$ . For the reverse, if $r \in / I$ , the multiplicatively closed set ${1, r, r^{2}, \dots}$ is disjoint from $I$ ; by a standard application of Zorn's lemma (proved in the Full proof set of 01.03.01), there is a prime $p \supseteq I$ with $r \in / p$ .

Exercise 8 (hard, symbolic).

Prove directly from the ACC that, in a Noetherian ring, every ideal admits a finite primary decomposition (the existence half of the Lasker–Noether theorem).

Hint

Call an ideal decomposable if it is a finite intersection of primary ideals. Show the set of non-decomposable ideals, if nonempty, has a maximal element $a$ ; derive a contradiction by writing $a = (a : x) \cap \dots$ after splitting on whether $a$ is primary.

Answer

Suppose the collection $N$ of non-decomposable ideals is nonempty; by condition (3), it has a maximal element $a$ . Then $a$ is not primary, so there exist $a, b$ with $ab \in a$ , $a \in / a$ , and $b^{n} \in / a$ for all $n$ ; in particular $b \in / a$ . The chain of colon ideals $a : (b) \subseteq a : (b^{2}) \subseteq \dots$ stabilises at some $a : (b^{n})$ by the ACC, and one checks $a = (a + (b^{n})) \cap (a : (b^{n}))$ . Both factors strictly contain $a$ , so by maximality each is decomposable, hence their intersection $a$ is decomposable — a contradiction. So $N$ is empty.

Lean formalization Intermediate+

lean_status: none is recorded because the project has not yet built the local bridge from the elementary statements here to the downstream consumers — the scheme spectrum in 04.02.01, the Dedekind-domain arithmetic of 21.02.07, and the local-field structure of 21.02.03. Mathlib itself already carries the full IsNoetherianRing hierarchy, ideal lattices, Localization and Localization.AtPrime, the Zariski topology on PrimeSpectrum, and primary-decomposition lemmas, so the missing work is naming and glue, not new infrastructure.

Advanced results Master

The three pillars — Noetherian rings, localisation, and primary decomposition — assemble into the working toolkit of algebraic geometry and number theory. We collect the results that make this toolkit load-bearing.

Lasker–Noether theorem. In a Noetherian ring every ideal admits a finite primary decomposition $I = q_{1} \cap \dots \cap q_{n}$ with each $q_{i}$ $p_{i}$ -primary. The radicals $p_{i}$ (the associated primes of $I$ ) are determined by $I$ up to reordering. Among them, the minimal (or isolated) primes — those minimal over $I$ — are intrinsic, and the corresponding primary components are unique; the embedded primes — the non-minimal associated primes — are not, and their components depend on the chosen decomposition ^{[Atiyah-Macdonald Ch. 4, Eisenbud Ch. 3]}. For $I = (x^{2}, x y)$ in $k [x, y]$ the minimal prime is $(x)$ and $(x, y)$ is embedded: this is the canonical picture of an embedded component.

Topological consequences. When $R$ is Noetherian, $Spec R$ is a Noetherian topological space: descending chains of closed sets stabilise. It decomposes uniquely into finitely many irreducible components, one for each minimal prime of $R$ . Every open cover of $Spec R$ has a finite subcover (quasi-compactness), and the closed points are dense in the closure of every point — a shadow of the dimension theory that algebraic geometry develops on top of this ^{[Atiyah-Macdonald Exercises 1.15, 6.6]}.

Nakayama's lemma. If $(R, m)$ is a local ring, $M$ a finitely generated $R$ -module, and $m M = M$ , then $M = 0$ . Equivalently, elements $m_{1}, \dots, m_{n} \in M$ generate $M$ iff their images generate the vector space $M / m M$ over the residue field $R / m$ ^{[Atiyah-Macdonald Proposition 2.6, Matsumura §2]}. This is the standard tool for lifting generators from the "special fibre," and it appears whenever a local argument must reduce a module to a vector space.

Krull's intersection theorem. In a Noetherian local ring $(R, m)$ , the powers of $m$ intersect in zero: $⋂_{n} m^{n} = 0$ . Combined with Nakayama this makes the $m$ -adic topology Hausdorff and underlies the Artin–Rees lemma and the dimension theory of local rings ^{[Atiyah-Macdonald Ch. 10, Matsumura §3]}.

Synthesis. The whole package — Noetherianity, localisation, primary decomposition, the Zariski topology on $Spec$ — is the foundational reason a commutative ring can be studied geometrically: prime ideals become the points of a space, localisation zooms into a point, and primary decomposition factors an ideal the way integers factor into primes. This is exactly the dictionary that lifts to affine schemes in 04.02.01, where $Spec R$ with its structure sheaf becomes a genuine geometric object; the local-ring construction generalises to the stalks of sheaves, the central insight being that local-to-global principles in geometry reduce to passing between a ring and its localisations. The bridge is that Noetherianity descends from rings to schemes and ascends to cohomology, and putting these together, the algebraic number theory of 21.02.07, the scheme theory of 04.02.01, and the module theory of 01.03.01 all reduce to the ideal arithmetic established here.

Full proof set Master

Proposition (Equivalence of the Noetherian conditions). For a commutative ring $R$ the following are equivalent: (1) the ACC on ideals; (2) every ideal is finitely generated; (3) every nonempty family of ideals has a maximal element.

Proof. $(1 \Rightarrow 2)$ Given an ideal $I$ , build a chain by adjoining generators: $I_{0} = 0$ , and $I_{n + 1} = I_{n} + (a_{n + 1})$ for some $a_{n + 1} \in I ∖ I_{n}$ if $I_{n} ⊊ I$ . By the ACC this chain stabilises at some $I_{N} = I$ ; then $I = (a_{1}, \dots, a_{N})$ is finitely generated.

$(2 \Rightarrow 1)$ Given an ascending chain $I_{1} \subseteq I_{2} \subseteq \dots$ , the union $I = ⋃_{n} I_{n}$ is an ideal. By (2), $I = (a_{1}, \dots, a_{k})$ ; each $a_{j}$ lies in some $I_{n_{j}}$ , so for $N = max_{j} n_{j}$ all generators already lie in $I_{N}$ , giving $I = I_{N}$ and the chain stabilises at $N$ .

$(1 \Rightarrow 3)$ If a nonempty family $F$ of ideals had no maximal element, pick $I_{1} \in F$ , then $I_{2} ⊋ I_{1}$ , then $I_{3} ⊋ I_{2}$ , and so on indefinitely — a strictly ascending chain violating the ACC. $(3 \Rightarrow 1)$ A chain $I_{1} \subseteq I_{2} \subseteq \dots$ is a family; by (3) it has a maximal element $I_{N}$ , which the chain cannot strictly exceed, so it stabilises at $N$ . $□$

Proposition (Prime correspondence under localisation). Let $S \subseteq R$ be multiplicatively closed. The assignment $p \mapsto S^{- 1} p$ is an order-preserving bijection between prime ideals of $R$ disjoint from $S$ and prime ideals of $S^{- 1} R$ , with inverse $q \mapsto ι^{- 1} (q)$ .

Proof. First note that $S^{- 1} p$ is prime when $p \cap S = \emptyset$ : if $(a / s) (b / t) = ab / s t \in S^{- 1} p$ , then $u (ab) \in p$ for some $u \in S$ , so $ab \in p$ (since $p \cap S = \emptyset$ means $u$ is not a zero-divisor mod $p$ ; more directly, $u \in / p$ and $p$ prime give $ab \in p$ ), hence $a \in p$ or $b \in p$ . The image is a proper ideal because $p \cap S = \emptyset$ prevents $1/1 \in S^{- 1} p$ .

Conversely, if $q ◃ S^{- 1} R$ is prime, then $ι^{- 1} (q)$ is a prime of $R$ disjoint from $S$ (disjointness holds because elements of $S$ map to units, which cannot lie in the proper ideal $q$ ). The two constructions are inverse: $ι^{- 1} (S^{- 1} p) = p$ follows from $a /1 = b / s$ iff $u (a s - b) = 0$ for some $u \in S$ , which for prime $p$ disjoint from $S$ forces $a \in p ⟺ b \in p$ ; and $S^{- 1} (ι^{- 1} (q)) = q$ because any $a / s \in q$ is $(a /1) (1/ s)$ with $a /1 \in q$ , i.e. $a \in ι^{- 1} (q)$ . Order preservation is immediate from the construction. $□$

Corollary. $Spec S^{- 1} R$ embeds as the open subset ${p \in Spec R : p \cap S = \emptyset}$ of $Spec R$ . For $S = R ∖ p$ this open set is the set of all primes contained in $p$ — the "neighbourhood of $p$ " in the Zariski topology, confirming that localising at $p$ zooms into the point $p$ .

Connections Master

Affine schemes and the Zariski topology 04.02.01. The spectrum $Spec R$ with its Zariski topology is the underlying space of an affine scheme; every construction in algebraic geometry starts from the ideal arithmetic developed here. Localisation produces the stalks of the structure sheaf, primary decomposition controls the scheme's embedded components, and Noetherianity of $R$ is exactly the finite-type condition that makes the scheme Noetherian.
Algebraic number theory and Dedekind domains 21.02.07. The ring of integers of a number field is a Dedekind domain — Noetherian, integrally closed, of Krull dimension one — in which every nonzero ideal factors uniquely into prime ideals. That factorisation is the number-theoretic shadow of primary decomposition, and unique factorisation of ideals is the theorem that repairs the failure of unique factorisation of elements in general number fields.
Rings and modules: the ascending chain condition 01.03.01. The module-theoretic ACC established in the prerequisite unit is the engine behind every chain argument here: a ring is Noetherian precisely when it is Noetherian as a module over itself, and the submodule chain arguments used to prove Hilbert's basis theorem and the Lasker–Noether theorem are the module ACC in disguise.
Local fields and $p$ -adic numbers 21.02.03. The $p$ -adic integers $Z_{p}$ and the field $Q_{p}$ arise by completing the local ring $Z_{(p)}$ ; they are the local-field endpoints of the localisation process. Every local-to-global principle in number theory passes through the local rings constructed in this unit before being lifted to the completed fields.

Historical & philosophical context Master

Commutative algebra crystallised over four decades around a single problem: understanding systems of polynomial equations through their ideals. Hilbert's 1890 paper Ueber die Theorie der algebraischen Formen proved that ideals in finitely generated polynomial rings over a field satisfy the finite-basis condition — the result stated as the basis theorem above — and in doing so introduced the chain-of-ideals viewpoint that replaced explicit computation with structural existence arguments ^{[Hilbert 1890]}. Lasker's 1905 Zur Theorie der moduln then showed that every polynomial ideal admits a primary decomposition, generalising integer factorisation to ideals.

Emmy Noether's 1921 paper Idealtheorie in Ringbereichen unified both results under the ascending chain condition: she proved that the ACC alone (equivalent to finite generation of every ideal) is enough to recover both Hilbert's theorem and primary decomposition, and the rings satisfying it now carry her name ^{[Noether 1921]}. This is the move that turned a collection of techniques into a theory: a single finiteness hypothesis, transparently stated, generates the whole edifice. Atiyah and Macdonald's 1969 Introduction to Commutative Algebra fixed the modern pedagogical sequence — rings, ideals, modules, localisation, primary decomposition, the spectrum — that this unit follows ^{[Atiyah-Macdonald 1969]}.

The philosophical shift is that geometry and algebra become two readings of the same object. A prime ideal is simultaneously an algebraic entity (the kernel of a map to an integral domain) and a geometric one (a point of a space); localisation is simultaneously a ring operation (inverting elements) and a geometric one (restricting to a neighbourhood). The whole of modern algebraic geometry is the working-out of this equivalence, which is why the spectrum and the local ring reappear in every chapter of §04.

Bibliography Master

@book{AtiyahMacdonald1969,
  author = {Atiyah, M. F. and Macdonald, I. G.},
  title = {Introduction to Commutative Algebra},
  publisher = {Addison-Wesley},
  year = {1969},
}

@article{Hilbert1890,
  author = {Hilbert, David},
  title = {Ueber die Theorie der algebraischen Formen},
  journal = {Mathematische Annalen},
  volume = {36},
  year = {1890},
}

@article{Lasker1905,
  author = {Lasker, Emanuel},
  title = {Zur Theorie der Moduln und Ideale},
  journal = {Mathematische Annalen},
  volume = {60},
  year = {1905},
}

@article{Noether1921,
  author = {Noether, Emmy},
  title = {Idealtheorie in Ringbereichen},
  journal = {Mathematische Annalen},
  volume = {83},
  year = {1921},
}

@book{Eisenbud1995,
  author = {Eisenbud, David},
  title = {Commutative Algebra with a View Toward Algebraic Geometry},
  publisher = {Springer GTM 150},
  year = {1995},
}

@book{Matsumura1989,
  author = {Matsumura, Hideyuki},
  title = {Commutative Ring Theory},
  publisher = {Cambridge University Press},
  year = {1989},
}

Prerequisites

01.03.01

Tier anchors

beginner: Intuition via Z and k[x] as Noetherian; ideals factored like integers (Atiyah-Macdonald 1969, Ch. 1–4 in plain English)
intermediate: Atiyah-Macdonald 1969, Introduction to Commutative Algebra, Ch. 1–7 (formal definitions and proofs)
master: Atiyah-Macdonald Ch. 1–7; Eisenbud 1995, Commutative Algebra with a View Toward Algebraic Geometry; Matsumura 1989, Commutative Ring Theory

References

Atiyah-Macdonald — Introduction to Commutative Algebra · Ch. 1 rings and ideals; Ch. 3 localisation; Ch. 4 primary decomposition; Ch. 6 chain conditions; Ch. 7 Noetherian rings
Eisenbud — Commutative Algebra with a View Toward Algebraic Geometry · Ch. 1 roots of commutative algebra; Ch. 2 fibre products; §1.6 Noetherian rings and Hilbert's basis theorem
Matsumura — Commutative Ring Theory · §2 prime spectra; §3 localisation; §6 Noetherian rings; §9 primary decomposition
Noether — Idealtheorie in Ringbereichen, Math. Ann. 83 (1921) · §1 chain conditions on ideals; the founding statement that ACC is equivalent to finite generation, with the decomposition theorem
Hilbert — Ueber die Theorie der algebraischen Formen, Math. Ann. 36 (1890) · §theorem that the polynomial ring in finitely many variables over a field or over Z satisfies the finite-basis condition
Lasker — Zur Theorie der moduln, Math. Ann. 60 (1905) · primary decomposition of polynomial ideals, the precursor to the Noetherian version

Estimated time

beginner: 20 min
intermediate: 55 min
master: 100 min