Title: Theory / Important / Math / Al gebra / Abstract Algebra - Dummit - Foote+14{ / 5 Commutative=Grobner
Tags: fromMath mathToDef fromImplication examplesSwitchExample1 mathToSee closed hasSeenConjDef list mid1516974248815 property toUnary

Name: Noetherian ring
Page: 656, 657, 657, https://math.stackexchange.com/questions/106122/subring-of-a-finitely-generated-noetherian-ring-need-not-be-noetherian
Section: 1
Chapter: 15
Index: 1, 3, 4
Kind: Prop, Theo, Cor
Extra: homo?
Under: quotient by ideal
Under2: homomorphic image
Under3: transforming it into the ring of polynomials
Condition3: we have a finite number of variables
Not4: Take \(K\) a ring
\(K[x,y]\) is Noetherian
Subring generated by \({xy^n\mid n\in\mathbb N}\) is not
Under4: subring
Under5: product
Condition5: finite
CounterExample5: take a product of rings whose ideal chain get longer
Under6: inverse of morphism?
Not7: ??
Under7: square matrix

Name: Commutative ring not Noetherian
Page: 657
Chapter: 15
Section: 1
Kind: Ex
Example: \(\mathbb Z[x_1,x_2,\dots]\).

Name: The ascending chain condition on \(R\)
Definition: There is no infinite ascending chain of ideals of \(R\).
Page: 656
Chapter: 15
Section: 1
Variables: \(R\) a commutative ring with \(0\ne 1\)
Kind: Def
Abbreviation: ACC on \(R\)

Name: The descending chain condition on \(R\)
Definition: "There is no infinite descending chain of ideals of \(R\)."
Page: 657
Chapter: 15
Section: 1
Variables: \(R\) a commutative ring with \(0\ne 1\)
Kind: Ex
Abbreviation: DCC on \(R\)

Name: A \(k\)-algebra homomorphism
Definition: A map \(\phi:R\to S\) such that \(\phi\) is the identity on \(k\).
Page: 657
Chapter: 15
Section: 1
Variables: \(k\) a field,
\(R,S\) be \(k\)-algebra, with \(0\ne 1\),
Index: (2)
Kind: Def

Smaller0: \(R\) is a finite-dimensional \(k\)-vector space
Greater0: \(R\) is finitely generated \(k\)-algebra
Page: 658, 657
Chapter: 15
Section: 1
Variables: \(k\) a field,
\(R\) a \(k\)-algebra
Kind: Ex,
Greater1: \(R\) satisfies both ACC and DCC
Connect Greater0: and

Name: \(k\)-algebra which is a finite-dimensional \(k\)-vector space
Page: 658
Chapter: 15
Section: 1
Variables: \(k\) a field,
Kind: Ex,
Example: \(k[x]/(p(x))\) with \(p\) not constant
k^n ?

Name: Affine \(n\)-space over \(k\)
Definition: The set of \(n\)-tuples of elements of \(k\).
Page: 658
Chapter: 15
Section: 1
Variables: \(k\) a field,
Notation: \(\mathbb A^n\)

Name: The coordinate ring of \(\mathbb A^n\)
Definition: The ring of polynomials in \(k\) with \(n\) variables, acting as \(k\)-valued \(n\)-ary functions.
Page: 658
Chapter: 15
Section: 1
Variables: \(k\) a field,
\(n\in\mathbb N\)
Notation: \(k[\mathbb A^n]\)

Smaller0: The set of solutions of \(f_i=g_i\)
Greater0: \(\mathcal Z(\{f_i-g_i\mid i\})\)
Page: 659
Chapter: 15
Section: 1
Variables: \(k\) a field,
\(n\in\mathbb N\),
\(f_1,\dots,f_m,g_1,\dots,g_m\in k[\mathbb A^n]\)
Kind: Def
Smaller to greater: is equal to
Hide middle: yes

Name: A linear algebraic set
Definition: The locus of a set of linear equations.
Page: 659
Chapter: 15
Section: 1
Variables: \(k\) a field,
\(n\in\mathbb N\),
Index: (3)
Kind: Ex
Example: Points
Example2: line
Example3: planes
Example4: hyperplanes

Smaller0: \(S\subseteq T\)
Greater0: \(\supseteq\)
Page: 659
Chapter: 15
Section: 1
Variables: \(k\) a field,
\(n\in\mathbb N\),
\(S,T\subseteq k[\mathbb A^n]\)
Index: (1)
Hide middle: yes
Prefix Greater: \(\mathcal Z(S)\)
Suffix Greater: \( \mathcal Z(T)\)

Smaller0: \(T= (S)\)
Greater0: \(=\)
Page: 659
Chapter: 15
Section: 1
Variables: \(k\) a field,
\(n\in\mathbb N\),
\(S,T\subseteq k[\mathbb A^n]\)
Index: (2)
Hide middle: yes
Prefix Greater: \(\mathcal Z(T)\)
Suffix Greater: \( \mathcal Z(S)\)

Name: The set of locus
Page: 659
Section: 1
Chapter: 15
Index: (3) (4)
Variables: \(k\) a field,
\(n\in\mathbb N\),
Extra: arbitrary union is not considered. Trivial since singletons are affine algebraic set
Under: intersections
Under2: product
Condition2: finite
Under4: union

Smaller0: \(\{fg\mid f\in S, g\in T\} \)
Greater0: \(\mathcal Z(S)\cup\mathcal Z(T)\)
Page: 659
Chapter: 15
Section: 1
Variables: \(k\) a field,
\(n\in\mathbb N\),
\(S,T\subseteq k[\mathbb A^n]\)
Index: (4)
Smaller to greater: =
Prefix Smaller: \(\mathcal Z(\)
Suffix Smaller: \()\)

Smaller0: \(S\cup T\)
Greater0: \(\mathcal Z(S)\cap\mathcal Z(T)\)
Page: 659
Chapter: 15
Section: 1
Variables: \(k\) a field,
\(n\in\mathbb N\),
\(S,T\subseteq k[\mathbb A^n]\)
Index: (3)
Smaller to greater: =
Prefix Smaller: \(\mathcal Z(\)
Suffix Smaller: \()\)

Q: \(\mathcal Z([...])=[...]\)
A: 1; \emptyset
Page: 659
Chapter: 15
Section: 1
Variables: \(k\) a field,
\(n\in\mathbb N\),
Index: (5)

Question: The maximal ideal of polynomials vanishing on every elements of \(A\) is
Answer: \(\mathcal I(A)\)
Page: 660
Chapter: 15
Section: 1

Question: \(\mathbf I(\{\vec a\})\) equals
Answer: \((x_1-a_1,\dots,x_n-a_n)\)
Page: 660
Chapter: 15
Section: 1
Variables: \(k\) a field,
\(\vec a\in\mathbb A^n\)

Smaller0: \(\mathcal I(\bigcup A_i) \)
Greater0: \(\bigcap\mathcal I(A_i) \)
Page: 661
Chapter: 15
Section: 1
Variables: \(k\) a field,
\(A_1\subseteq A_2\subseteq \mathbb A^n\)
Index: 7
Kind: Exemple
Smaller to greater: =

Smaller0: \(\emptyset\)
Greater0: \(k[x_1,\dots,x_n]\)
Page: 660
Chapter: 15
Section: 1
Variables: \(k\) a field,
Index: 6
Kind: Exemple
Smaller to greater: =
Prefix Smaller: \(\mathcal I(\)
Suffix Smaller: \()\)

Smaller0: \(\mathbb A^n\)
Greater0: \(0\)
Page: 660
Chapter: 15
Section: 1
Variables: \(k\) a field,
Index: 8
Kind: Exemple
Smaller to greater: =
Assuming: \(k\) infinite
Prefix Smaller: \(\mathcal I(\)
Suffix Smaller: \()\)

Smaller0: \(\{x_i-a_i\mid i\} \)
Greater0: \(\{\vec a\} \)
Page: 658
Chapter: 15
Section: 1
Variables: \(k\) a field,
\(\vec a\in \mathbb A^n\)
Index: 2
Kind: Example
Smaller to greater: =
Prefix Smaller: \(\mathbf Z(\)
Suffix Smaller: \()\)

Question: \(\mathcal Z\circ\mathcal I\) has the property
Answer: to send a set to a superset
Page: 661
Chapter: 15
Section: 1
Variables: \(k\) a field,
Index: 9
Kind: Example

Question: \(\mathcal I\circ\mathcal Z\) has the property
Answer: To send an ideal to a superset
Page: 661
Chapter: 15
Section: 1
Variables: \(k\) a field,
Index: 9
Kind: Example

Question: \(\mathcal I\circ\mathcal Z\circ \mathcal I\)
Answer: equals \(\mathcal I\)
Page: 661
Chapter: 15
Section: 1
Variables: \(k\) a field,
Context: Field theory
Index: 9
Kind: Example

Question: \(\mathcal Z\circ\mathcal I\circ\mathcal Z\) is
Answer: \(\mathcal Z\)
Page: 661
Chapter: 15
Section: 1
Variables: \(k\) a field,
Index: 9
Kind: Example

Name: The coordinate ring of \(V\)
Definition: \(k[\mathbb A^n]/\mathcal I(V)\)
Page: 661
Chapter: 15
Section: 1
Variables: \(k\) a field,
\(V\subseteq \mathbb A^n\)
Notation: \(k[V]\)
Kind: Def

Smaller0: \(f=g\) in \(k[V]\)
Greater0: \(f(\vec x)=g(\vec x)\) for each \(\vec x\in V\)
Page: 661
Chapter: 15
Section: 1
Variables: \(k\) a field,
\(V\subseteq \mathbb A^n\)
\(f,g\in k[x_1,\dots,x_n]\)
Smaller to greater: iff

Name: A morphism of algebraic set
Definition: A function \((a_1,\dots,a_n)\mapsto(\phi_1(a_1,\dots,a_n),\dots,\phi_m(a_1,\dots,a_n))\) with \(\phi_i\in[x_1,\dots,x_n]\)
Page: 662
Chapter: 15
Section: 1
Variables: \(k\) a field,
\(V_i\subseteq \mathbb A^{n_i}\) affine algebraic sets
Name2: A polynomial map of algebraic set
Name3: A regular map of algebraic set

Name: An isomorphism of algebraic set
Definition: A morphism of algebraic set with a two-sided inverse morphism of algebraic set
Page: 662
Chapter: 15
Section: 1
Variables: \(k\) a field,
\(V_i\subseteq \mathbb A^{n_i}\) affine algebraic sets

Definition: \(\phi\mapsto (f\mapsto f\circ\phi)\) for each \(n_1,n_2\in\mathbb N\), \(V_i\in\mathbb A^i\) affine algebraic sets, and \(\phi:K[V_1]\to K[V_2]\)
Page: 663
Variables: \(k\) a field
Notation: \(\tilde\cdot\) on morphism of algebraic set
Index: Theo
Kind: 6 (3)
Monotonic: contravariant morphism
Jective: bijection

Smaller0: a \(k\)-algebra isomorphism
Greater0: \(\phi\) is a ring-isomorphism
Page: 663
Variables: \(k\) a field
\(n_1,n_2\in\mathbb N\),
\(V_i\subseteq\mathbb A^{n_i}\) affine algebraic sets, and
\(\phi:K[V_1]\to K[V_2]\)
Index: 6 (4)
Kind: Theo
Smaller to greater: iff
Prefix Smaller: \(\tilde\phi\) is

Smaller0: for every \(f\in k[V_2]\), \(f\circ\phi\in k[V_1]\)
Greater0: \(\phi\) is an affine-morphism
Page: 663
Variables: \(R\) a commutative ring with \(1\ne 0\),
\(n_1,n_2\in\mathbb N\),
\(V_i\subseteq\mathbb A^i\) affine algebraic sets and
\(\phi:V_1\to V_2\) a map
Index: 7
Kind: Cor
Smaller to greater: iff

Smaller0: \(\tilde\phi^{-1}(\mathcal I(\{v\}))=\mathcal I(\{w\})\)
Greater0: \(\phi(v)=w\)
Page: 664
Variables: \(R\) a commutative ring with \(1\ne 0\),
\(n_1,n_2\in\mathbb N\),
\(V_i\subseteq\mathbb A^i\) affine algebraic sets
\(\phi:V_1\to V_2\), a morphism
\(v_i\in V_i\)
Index: 7
Kind: Cor
Smaller to greater: iff

Question: The number of \(\psi\) such that \(\phi=\tilde\psi\) is
Answer: exactly one
Page: 663
Variables: \(R\) a commutative ring with \(1\ne 0\),
\(n_1,n_2\in\mathbb N\),
\(V_i\in\mathbb A^i\) affine algebraic sets and
\(\phi:K[V_1]\to K[V_2]\),
Index: 6
Kind: Theo
Assuming: \(\phi\) is a \(k\)-algebra isomorphism

Question: Blocked because of
Answer: Grobner
Page: 660
Chapter: 15
Section: 1
Index: 6
Kind: Ex
Assuming: \(\phi\) is a \(k\)-algebra isomorphism

Question: The intuition of \(f\) in \(k[V]\)
Answer: The restriction of \(f\) to \(V\)
Page: 661
Chapter: 15
Section: 1
Variables: \(k\) a field,
\(V\subseteq \mathbb A^{n}\)
\(f\in k[x_1,\dots,x_n]\)

Name: An affine algebraic set
Definition: a finite intersection of hypersurfaces in \(\mathbb A^n\).
Page: 659, 660
Chapter: 15
Section: 1
Variables: \(k\) a field,
\(n\in\mathbb N\),
Kind: Def,

Name: A Noetherian commutative ring
Definition: It satisfies the ACC,
Page: 656, 657e
Chapter: 15
Section: 1
Index: , 2
Kind: Def, Theo, Ex
Definition2: Every nonempty set of ideals of \(R\) contains a maximal element under inclusion
Example: The PID
Conjdef: equivalently
Definition3: Every ideal of \(R\) is finitely generated
DefType: A commutative ring \(R\) with \(1\ne 0\)

Name: The locus of \(S\) in \(\mathbb A^n\)
Definition: The set of points where all functions in \(S\) are simultaneously \(0\).
Page: 658, 659
Chapter: 15
Section: 1
Variables: \(k\) a field,
\(n\in\mathbb N\),
\(S\subseteq k[\mathbb A^n]\)
Notation: \(\mathcal Z(S)\)
Index: , (1), (3)
Kind: , Ex

Smaller0: \(s=\{f\}\)
Greater0: the set of roots of \(f\)
Page: 658, 659
Chapter: 15
Section: 1
Variables: \(k\) a field,
\(n\in\mathbb N\),
\(S\subseteq k[\mathbb A^n]\)
Kind: , Ex
Prefix Greater: \(\mathcal Z(S)\) is

Question: Example of finitely generated infinite dimensional space ?
Answer: \(k[x]\)
Page: 658, 657
Chapter: 15
Section: 1
Variables: \(k\) a field,
\(R\) a \(k\)-algebra
Kind: Ex,

Smaller0: \(0\)
Greater0: \(\mathbb A^n\)
Page: 659
Chapter: 15
Section: 1
Variables: \(k\) a field,
\(n\in\mathbb N\),
Index: (5)
Smaller to greater: =
Hide middle: yes
Prefix Smaller: \(\mathcal Z(\)
Suffix Smaller: \()\)

Name: A finitely generated \(k\)-algebra
Definition: A commutative ring \(R\) with \(0\ne 1\) such that \(R\) is generated as a ring by \(k\) together with some finite set of elements of \(R\)
Page: 657, 658
Chapter: 15
Section: 1
Variables: \(k\) a field
Index: (1), 5
Kind: Def, Cor
Definition2: A ring \(R\) such that there is some surjective \(k\)-algebra homomorphism from \(k[x_1,\dots,x_n]\) to \(R\).
Extra: «by \(k\)» ?

Name: A hyper-surface in \(\mathbb A^n\)
Definition: The locus of a singleton.
Page: 659
Chapter: 15
Section: 1
Variables: \(k\) a field,
\(n\in\mathbb N\),
Index: (4)
Kind: Ex
Example: parabola, \(y-x^2\)
Example2: quadratic surfaces, \(\sum_{i=0}^2\sum_{j=0}^2 a_{i,j}x^iy^j\)
Example3: circle, \(x^2+y^2-1\),
Example4: \(xy-1\)hypebola