Title: Theory / Important / Math / Al gebra / Abstract Algebra - Dummit - Foote+14{ / 2 Rings
Tags: fromMath mathToDef fromImplication hasSeenConjDef examplesSwitchExample1 counterExample defToAlgebra exampleToQuestion mathToSee questionExample TypToQuestion closed mid1516974248815 récupéré toBinaryRelation automaticImplies checked conjDefToType defToAlgebra2 setAdditionMultiplication setMultiplication toAlgebra toUnary typToBinary

Name: A ring
Definition: such that \((R,+)\) is an abelian group,
Page: 223
Chapter: 7
Section: 1
Index: (1)
Kind: Def
Definition2: \(\times\) is associative and
Definition3: the distributive laws hold in \(R\).
DefType: A 3-tuple of the form \((R,+,\times)\) with

Name: Commutative field
Definition: A field \((R,+,\times)\) where \(\times\) is commutative.
Page: 223
Chapter: 7
Section: 1
Index: (2)
Kind: Def

Name: An identity
Definition: An element \(x\in R\) such that \(xy=yx=y\) for all \(y\in R\).
Page: 223
Chapter: 7
Section: 1
Variables: \((R,+,\times)\) a field
Index: (3)
Kind: Def

Smaller0: \(\times\) admits an identity and \(\times\) is distributive over \(+\)
Greater0: \(+\) is Abelian
Page: 223
Chapter: 7
Section: 1
Variables: \((R,+,\times)\) a 3-tuple

Page: 224
Chapter: 7
Section: 1
Name: Zero ring
Set: \(\{0\}\)
*: Trivial
Kind: Ex
+: 0
Commutative: y

Name: A trivial ring
Definition: \((R,+)\) a group
Page: 224
Chapter: 7
Section: 1
Kind: Ex
Definition2: \(a\times b\mapsto 0\).
Conjdef: \(\left(R,+,\times\right)\) with

Name: Rings \(R\) with identity and whose multiplication (on \(R\setminus \{0\}\)) is not a group
Page: 224
Chapter: 7
Section: 1
Index: 2 4
Kind: Ex
Example: \(\mathbb Z\)
Example2: \(\mathbb Z/n\mathbb Z\)

Name: Rings with identity and whose \(\times\) is a group
Page: 224
Chapter: 7
Section: 1
Index: 3
Kind: Ex
Example: \(\mathbb Q,\mathbb R, \mathbb C\)

Name: A division ring, noncommutative with identity
Page: 224
Chapter: 7
Section: 1
Index: 5
Kind: ex
Example: \(\mathbb H\), the (real) Hamilton Quaternions

Question: \(0a=a0=\)
Answer: \(0\).
Page: 226
Chapter: 7
Section: 1
Variables: \((R,+,\times)\) a ring.
Kind: Prop

Question: \((-a)b=a(-b)=\)
Answer: \(-(ab)\).
Page: 226
Chapter: 7
Section: 1
Variables: \((R,+,\times)\) a ring.
Index: 1 (2)
Kind: Prop

Question: \((-a)(-b)=\)
Answer: \(ab\).
Page: 226
Chapter: 7
Section: 1
Variables: \((R,+,\times)\) a ring.
Index: 1(3)
Kind: Prop

Name: Zero divisor
Definition: \(a\ne0\) such that there exists \(b\ne 0\) such that \(ab=0\) or \(ba=0\).
Page: 226, 237
Chapter: 7, 7
Section: 1, 2
Variables: \((R,+,\times)\)
Index: , 1
Kind: Def, ex
Example: \((1-g)(\sum_{i=1}^{\text{order of }g-1}g^i)\), for \(g\ne 1\)
Example2: any divisor of \(n\)
Example3: \(\emptyset\)
Applied to: Let \(R\) a commutative ring with identity \(1\ne 0\) and \(G\) a finite multiplicative group.
Example4: the set of function whose kernel is not empty
Applied to3: \(\mathbb Z\)
Applied to2: \(\mathbb Z/n\mathbb Z\)
Applied to4: function with real values

Name: A unit of \(R\)
Definition: An element \(a\in R\) such that there exists \(b\) such that \(ab=1\) or \(ba=1\) and \(1\) exists.
Page: 226, 227
Chapter: 7
Section: 1
Variables: \((R,+,\times)\) a ring
Index: , (3)
Kind: def, ex
Example: functions with an empty kernel
Example2: \(\pm 1\)
Example3: Elements coprime with \(n\)
Applied to: Function with real value
Applied to3: \(\mathbb Z/n\mathbb Z\)
Applied to2: \(\mathbb Z\)

Definition: whose set contains the units
Page: 226
Chapter: 7
Section: 1
Variables: \((R,+,\times)\) a ring
Notation: \(R^\times\)
Kind: def
DefType: The subgroup of \(R\)

Smaller0: \(a\) is a unit
Greater0: \(a\) is not a 0 divisor
Page: 226
Chapter: 7
Section: 1
Variables: \((R,+,\times)\) a ring
\(a\in R\)
Kind: Def
Counterexample: 2 in \(\mathbb Z\)

Smaller0: \(R\) is a field
Greater0: 0
Page: 226
Chapter: 7
Section: 1
Variables: \((R,+,\times)\) a ring
\(a\in R\)
Prefix Greater: the number of zero divisors of \(R\) is
Counterexample: \(\mathbb Z\)

Question: \(\mathbb Z^\times= \)
Answer: \(\{+1,-1\} \).
Page: 226
Chapter: 7
Section: 1
Index: (1)
Kind: Ex

Question: \((\mathbb Z/n\mathbb Z)^\times= \)
Answer: \(\{\overline u\mid \text{$u$ and $n$ are relatively prime} \} \).
Page: 226
Chapter: 7
Section: 1
Variables: \(n\in N\).
Index: (2)
Kind: Ex

Question: The zero divisors of \(\mathbb Z/n\mathbb Z\) are
Answer: \(\{\overline u\mid \text{$u$ and $n$ are not relatively prime} \} \).
Page: 226
Chapter: 7
Section: 1
Variables: \(n\in N\).
Index: (2)
Kind: Ex

Name: An integral domain
Definition: A commutative ring, such that, \(1\ne 0\) and which has no zero divisors.
Page: 228
Chapter: 7
Section: 1
Variables: \((R,+,\times)\) a ring.
Kind: Def
French: Anneau integre

Smaller0: \(a\) is not a zero divisor
Greater0: \(ab=ac\)
Page: 228
Chapter: 7
Section: 1
Variables: \((R,+,\times)\) a ring, \(a,b,c\in R\).
Index: 2
Kind: Prop
Greater1: \(b=c\)
Connect Greater0: iff

Smaller0: \(R\) finite
Greater0: \(R\) is an integral domain
Page: 228
Chapter: 7
Section: 1
Variables: \((R,+,\times)\) a ring.
Index: 3
Kind: Cor
Hide middle: yes
Greater1: \(R\) is a field
Connect Greater0: iff
Counterexample: Integral domain which is not a field: \(\mathbb Z\)

Page: 228
Definition: \(R\subseteq S\),
Chapter: 7
Section: 1
Name: The subring relation
Kind: Def
Transitive: Transitive
Definition2: \(+_R= \left(+_S\right)_{\mid R}\) and \(\times_R= \left(\times_S\right)_{\mid R}\)
Symmetry: The relation between pairs of rings \((R,+_R,\times _R)\) and \((S,+_S,\times _S)\) such that

Smaller0: \(D\equiv 1\mod 4\)
Greater0: \(\mathbb Z\left[\frac{1+\sqrt D}{2}\right]\)
Page: 229
Chapter: 7
Section: 1
Variables: \(D\in\mathbb Z\)
Kind: Ex
Hide middle: yes
Assuming: \(D\) is square free
Greater1: a subring of \(\mathbb Q[\sqrt D]\)
Connect Greater0: is

Name: The ring of integers in the quadratic field \(\mathbb Q(\sqrt D)\)
Definition: \(\mathbb Z\left[\frac{1+\sqrt D}2\right]\) if \(D\equiv 1\mod 4\)
Page: 229
Chapter: 7
Section: 1
Variables: \(D\in\mathbb Z\)
Notation: \(\mathcal O\)
Kind: Ex
Definition2: \(\mathbb Z\left[\sqrt D\right]\) if \(D\not\equiv 1\mod 4\)
Assuming: \(D\) square free

Name: A Gaussian integer
Definition: An integer of the form \(m+in\) with \(m,n\in\mathbb N\)
Page: 229
Chapter: 7
Section: 1
Kind: Ex

Name: The norm of \(a+b\sqrt D\in\mathbb Q[\sqrt D]\)
Definition: \((a+b\sqrt D)(a-b\sqrt D)=a^2-Db^2\).
Page: 229
Chapter: 7
Section: 1
Variables: \(D\) square free.
Kind: Ex
Abbreviation: \(N(a+b\sqrt D)\)

Smaller0: \(N(\alpha\beta)\)
Greater0: \(N(\alpha)N(\beta)\)
Page: 230
Chapter: 7
Section: 1
Variables: \(D\) square free
\(\alpha,\beta\in\mathcal O\)
Smaller to greater: =
Hide middle: yes

Definition: \((a+b\omega)(a+b\overline\omega)=\left\{\begin{array}{ll}a^2+ab+\frac{1-D}4b^2&\text{if $D\equiv 1\mod 4$}\\a-Db^2&\text{otherwise} \end{array}\right. \).
Page: 230
Chapter: 7
Section: 1
Variables: \(D\) square free, let \(a,b\in\mathbb Z\)
Notation: \(N(a+b\omega)\)

Page: 230
Chapter: 7
Section: 1
Variables: \(D\) square free,
\(a,b\in\mathbb Z\)
Notation: \(N(a+b\omega)\)
Typ: Integer

Smaller0: \(N(a+b\omega)=\pm1\)
Greater0: \(a+b\omega\) is a unit
Page: 230
Chapter: 7
Section: 1
Variables: \(D\) square free
\(a,b\in\mathbb Z\)
Smaller to greater: iff
Hide middle: yes

Question: The units of \(\mathbb Z\left[j\right]\) with \(j=\frac{1+\sqrt {-3} }2\) are
Answer: \(\{\pm1,\pm j,\pm j^2\}\).
Page: 230
Chapter: 7
Section: 1

Question: The units of \(\mathbb Z\left[\sqrt2\right]\) are
Answer: \(\{\pm(1+\sqrt 2)^n\mid n\in\mathbb Z\}\).
Page: 230
Chapter: 7
Section: 1
Extra: Checked
Construction: Norm is 1

Name: A polynomial
Definition: An element of the form \(\sum_{i=0}^n a_ix^i\) where \(x\) is an indeterminate, \(n\ge-1\), \(a_i\in R\) and \(a_0\ne 0\).
Page: 234
Chapter: 7
Section: 2
Variables: \(R\) a ring with identity

Name: Leading coefficient
Definition: \(a_n\)
Page: 234
Chapter: 7
Section: 2
Variables: \(R\) a ring with identity
 \(\sum_{i=0}^n a_ix^i\)

Name: Leading term
Definition: \(a_nx^n\)
Page: 234
Chapter: 7
Section: 2
Variables: \(R\) a ring with identity
\(\sum_{i=0}^n a_ix^i\)

Name: degree
Definition: \(n\)
Page: 234
Chapter: 7
Section: 2
Variables: \(R\) a ring with identity and \(\sum_{i=0}^n a_ix^i\) with \(a_n\ne 0\)

Name: Monic polynomial
Definition: A polynomial of the form \(p(x)=x^n+\sum_{i=0}^{n-1} a_ix^i\).
Page: 234
Chapter: 7
Section: 2
Variables: \(R\) a ring with identity.

Name: Perfect square in \(\mathbb Z/2\mathbb Z[x]\) which is not a perfect square in \(\mathbb Z[x]\)
Page: 235
Chapter: 7
Section: 2
Example: \(x^2+1\)

Smaller0: \(p\) is unit in \(R[x]\)
Greater0: \(p\) is unit in \(R\)
Page: 235
Chapter: 7
Section: 2
Variables: \(R\) a ring with unit and
\(p\in R[x]\)
Index: 4 (2)
Kind: Prop
Smaller to greater: iff
Assuming: \(R\) is an integral domain

Name: Integral domain
Page: 235
Section: 2
Chapter: 7
Index: 4 (3)
Kind: Prop
Under: transforming the ring into an algebra
Not2: not
Under2: product
CounterExample2: \(\mathbb Z\times\mathbb Z\) has \((0,1)(1,0)=(0,0)\)

Definition: of \(n\times n\) matrices with entries from \(R\).
Page: 235
Chapter: 7
Section: 2
Variables: \(R\) a ring.
Notation: \(M_n(R)\)
DefType: The ring

Name: \((a_{ij})+(b_{ij})\)
Definition: \((a_{ij}+b_{ij})\)
Page: 235
Chapter: 7
Section: 2
Variables: \(R\) a ring and \((a_{ij}),(b_{ij})\in M_n(R)\).

Name: The \(i,j\)-th entry of \((a_{ij})(b_{ij})\)
Definition: \(\sum_{k=1}^n a_{ik}b_{kj}\).
Page: 235
Chapter: 7
Section: 2
Variables: \(R\) a ring and
\((a_{ij}),(b_{ij})\in M_n(R)\).

Smaller0: commutative
Greater0: \(R\) is trivial
Page: 235
Chapter: 7
Section: 2
Variables: \(R\) a ring.
Greater1: \(n\le 1\)
Connect Greater0: or
Extra: n=1, R com
Prefix Smaller: \(M_n(R)\) is

Name: A scalar matrice
Definition: A matrix of the form  [$]
\begin{pmatrix}
a & 0 & \dots &0& 0\\
0 & a & \dots &0& 0\\
\vdots & \vdots&\ddots& \vdots&\vdots\\
0 & 0 & \dots &a& 0\\
0&0&\dots&0&a
\end{pmatrix}[/$] for some \(a\in R\)
Page: 236
Chapter: 7
Section: 2
Variables: \(R\) a ring

Name: The set of scalar matrice in \(M_n(R)\)
Page: 236
Chapter: 7
Section: 2
Variables: \(R\) a ring.
Typ: A subring of \(M_n(R)\)
Property: Isomorphic to \(R\)

Name: The identity matrix
Definition: \((a_{ij})\) such that \(a_{ij}\) is \(1\) if \(i=j\) and \(0\) otherwise.
Page: 236
Chapter: 7
Section: 2
Variables: \(R\) a ring.

Name: The general linear group of degree \(n\)
Definition: The set of units of \(M_n(R)\).
Page: 236
Chapter: 7
Section: 2
Variables: \(R\) a ring
\(n\in\mathbb N\)
Notation: \(GL_n(R)\)

Name: Subring relations
Page: 237
Section: 2
Chapter: 7
Under: \(R\mapsto M_n(R)\)
Under2: \(R\mapsto RG\) for \(G\) a group
Under3: product
Under4: \(R\mapsto R[X]\)

Name: The set of upper triangular matrices
Page: 236
Chapter: 7
Section: 2
Variables: \(R\) a ring and
\(n\in\mathbb N\)
Typ: A subring of \(M_n(R)\)

Page: 236, 841
Chapter: 7, 18
Section: 2, 1
Name: The group ring of \(G\) with coefficient in \(R\)
Set: set of formal sums \(\sum_{g\in G}a_gg\), with \(a_g\in R\).
Variables: \(R\) a commutative ring with identity \(1\ne 0\)
\(G\) a finite multiplicative group.
Notation: \(RG\)
*: \(\left(\sum_{g\in G}a_g,\sum_{g\in G}b_g\right)\mapsto\sum_{g\in G}\left(\sum_{g_1g_2=g}(a_{g_1}b_{g_2})\right)g\).
Kind: ,
Index: 2
+: \(\left(\sum_{g\in G}a_g,\sum_{g\in G}b_g\right)\mapsto\sum_{g\in G}(a_g+b_g)g\).
0: \(0\) (empty sum)
1: \(1_G1_R\)
Zero divisors: \((1-g)(\sum_{i=0}^{\text{ord} g}{g^i})=1-g^{\text{ord} g}=1-1=0\)
Typ: The ring
An \(R\)-module
An \(R\)-algebra,

Definition: \(a1_g\).
Page: 236
Chapter: 7
Section: 2
Variables: \(R\) a commutative ring with identity \(1\ne 0\) and \(G\) a finite multiplicative group. Let \(a\in R\)
Context: In \(RG\),
Notation: \(a\)

Definition: \(1_Rg\).
Page: 236
Chapter: 7
Section: 2
Variables: \(R\) a commutative ring with identity \(1\ne 0\) and
\(G\) a finite multiplicative group.
\(g\in G\)
Context: \(RG\)
Notation: \(g\)

Name: Subring of \(RG\)
Page: 237
Chapter: 7
Section: 2
Variables: \(R\) a commutative ring with identity \(1\ne 0\)
\(G\) a finite multiplicative group.
Example: The set of elements of \(RG\) whose sum is zero
Example2: \(R\)

Name: A ring homomorphism
Definition: A function from ring to ring which commutes with addition and multiplication
Page: 239
Chapter: 7
Section: 3
Index: (1)
Kind: Def

Name: A ring isomorphism
Definition: A bijective ring homomorphism.
Page: 239
Chapter: 7
Section: 3
Index: (3)
Kind: Def

Name: The kernel of \(\phi\)
Definition: \(\phi^{-1}(0_S)\).
Page: 239-241
Chapter: 7
Section: 3
Variables: \(R,S\) rings and \(\phi:R\to S\) a ring homomorphism
Index: (2)
Kind: Def
Abbreviation: ker \(\phi\)
Typ: An ideal of \(R\)

Page: 240
Definition: There exists an isomorphism from \(R\) to \(S\).
Chapter: 7
Section: 3
Variables: \(R,S\) rings.
Notation: \(R\cong S\)

Name: Group homomorphism which is not a ring homomorphism
Page: 240
Chapter: 7
Section: 3
Index: (2)
Kind: Ex
Example: [$]\defFun{\mathbb Z}{\mathbb Z}{n}{2n}[/$]

Page: 240
Chapter: 7
Section: 3
Variables: \(R,S\) rings
\(\phi:R\to S\) an homomorphism
Notation: \(\phi(R)\)
Index: 5 (1)
Kind: Prop
Typ: A subring of \(S\)

Name: A left ideal of \(R\)
Definition: A subring \(I\) of \(R\) such that \(RI=I\).
Page: 242,
https://fr.wikipedia.org/wiki/Id%C3%A9al
Chapter: 7
Section: 3
Variables: \(R\) a ring
Index: (2)
Kind: Def
Definition2: A sub module of \(S\) for the left-module of \(S\) on \(S\)

Name: An ideal of \(R\)
Definition: A left and right ideal of \(R\).
Page: 242
Chapter: 7
Section: 3
Variables: \(R\) a ring.
Index: (3), 7(2)
Kind: Def, ,
Definition2: The kernel of a ring homomorphism
Definition3: A subset \(I\subseteq R\) such that \(R/I\) is a ring, with addition and multiplication correctly defined
Denoted by: \(I\)

Definition: \(\left(\{r+I\mid r\in R\}, ((r+I),(s+I))\mapsto (r+s)+I, ((r+I),(s+I))\mapsto (rs)+I, 0+I)\right)\)
Page: 243
Chapter: 7
Section: 3
Variables: \(R\) a ring,
\(I\) an ideal of \(R\)
Notation: \(R/I\)
Kind: Def

Name: First isomorphism theorem for rings
Definition: \(R/\text{ker }\phi\) is isomorphic as a ring to \(\phi(R)\).
Page: 243
Chapter: 7
Section: 3
Variables: \(R,S\) rings and \(\phi:R\to S\) a ring homomorphism
Kind: 7(1)

Definition: [$]\defFun{R}{R/I}{r}{r+I}[/$].
Page: 243
Section: 3
Chapter: 7
Variables: \(R\) a ring
\(I\subseteq R\) an ideal of \(R\)
Name: The natural projection
Kind: 7(2)

Name: Trivial ideal
Definition: \(\{0\}\)
Page: 243
Chapter: 7
Section: 3
Variables: \(R\) a ring.
Index: (1)
Kind: Ex

Name: A proper ideal
Definition: An ideal of \(R\), distinct from \(R\).
Page: 243
Chapter: 7
Section: 3
Variables: \(R\) a ring
Index: (1)
Kind: Ex

Question: The ideals of \(\mathbb Z\) are
Answer: \(n\mathbb Z\) for \(n\in\mathbb N\).
Page: 243
Chapter: 7
Section: 3
Index: (2)
Kind: Ex

Name: Center of \(R\)
Definition: \(\{z\in R\mid zr=rz, \forall r\in R\}\).
Page: 231
Chapter: 7
Section: 1
Variables: \(R\) a ring.
Kind: Exercice

Name: A nilpotent element
Definition: An element \(x\in R\) such that \(x^m=0\) for some m.
Page: 231
Chapter: 7
Section: 1
Variables: \(R\) a ring,
Index: 13
Kind: Ex

Name: A Boolean ring
Definition: A ring such that \(a^2=a\) for each element \(a\).
Page: 231
Chapter: 7
Section: 1
Index: 15
Kind: Ex

Name: formal power series
Definition: Elements: of the form \(\sum_{i=0}^\infty a_ix^i\)
Page: 238
Chapter: 7
Section: 2
Variables: \(R\) a ring
Notation: [$]R\llbracket{}x
\rrbracket[/$]
Index: 3
Kind: Ex
Definition2: Pointwise addition
Conjdef: The ring
Definition3: Multiplication \( (\sum_i a_i x^i,\sum_i b_i x^i)\mapsto \sum_i \sum_{j=0}^i (a_jb_{i-j}) x^i\).

Name: The set of formal Laurent series with coefficients from \(F\)
Page: 238
Chapter: 7
Section: 2
Variables: \(F\) a field
Notation: \(F((x))\)
Index: 5
Kind: Ex

Name: A formal Laurent series with coefficients from \(F\)
Definition: Elements \(\{\sum_{i=N}^{\infty} a_i x^i\mid n\in\mathbb Z, a_i\in N\}\)
Page: 238
Chapter: 7
Section: 2
Variables: \(F\) a field
Index: 5
Kind: Ex
Definition2: sum \(\{\sum_{i=N}^{\infty} a_i x^i\mid n\in\mathbb Z, a_i\in N\},(\sum_{i=N}^{\infty} a_i x^i,\sum_{i=N'}^{\infty} b_i x^i)\mapsto \sum_{i=\min(N,N')}^{\infty} (a_i+b_i) x^i\)
Conjdef: The ring
Definition3: Product \((\sum_{i=N}^{\infty} a_i x^i,\sum_{i=N'}^{\infty} b_i x^i)\mapsto \sum_{i=N+N'} \sum_{j=0}^i (a_jb_{i-j}) x^i\)

Question: The quotient of \(\mathbb Z[x]\) by the ideal of polynomials whose monomials degree is at least 2
Answer: is isomorphic to the set of polynomials of degree at most one, with \(x^2=0\)
Page: 244
Chapter: 7
Section: 3
Kind: Ex

Definition: \(f\mapsto f(c)\)
Page: 244
Section: 3
Chapter: 7
Variables: \(A\) a ring,
\(X\) a nonempty set,
\(R=A^X\)
\(c\in X\)
Notation: \(E_c\)
Name: Evaluation at \(c\)
Index: Ex
Kind: 4
Morphism: ring homomorphism
Jective: Surjective

Question: \(M_n(R/J)\) is
Answer: an ideal of \(M_n(R)\)
Page: 244
Chapter: 7
Section: 3
Variables: \(n\in\mathbb N\),
\(R\) a ring,
\(J\) an ideal of \(R\)
Index: 6
Kind: Ex

Smaller0: \(M_n(R)/M_n(J)\)
Greater0: \(M_n(R/J)\)
Page: 244
Chapter: 7
Section: 3
Variables: \(n\in\mathbb N\),
\(R\) a ring,
\(J\) an ideal of \(R\)
Index: 6
Kind: Ex
Smaller to greater: is isomorphic to

Name: The augmentation map
Definition: \(\sum a_ig_i\mapsto\sum a_i\)
Page: 245
Chapter: 7
Section: 3
Variables: \(R\) a ring,
\(G=\{g_1,\dots,g_n\}\) a finite group
Kind: Ex
Typ: Ring homomorphism

Name: The augmentation ideal
Definition: The kernel of \(RG\) by its augmentation map
Page: 245
Chapter: 7
Section: 3
Variables: \(R\) a ring,
\(G=\{g_1,\dots,g_n\}\) a finite group
Kind: Ex
Definition2: \(\{\sum_{i=1}^na_ig_i\mid a_i\in R, \sum a_i=0_R\}\)
Denoted by: \(I\)

Question: The quotient of \(RG\) by its augmentation ideal is
Answer: Isomorphic to \(R\)
Page: 245
Chapter: 7
Section: 3
Variables: \(R\) a ring,
\(G=\{g_1,\dots,g_n\}\) a finite group
Kind: Ex

Question: Examples of left-sided ideal which is not a right-sided ideal
Answer: Given \(R\) a ring with identity, \(1\ne 0\), and \(n\in\mathbb N\), the ideal \(L_j\) of \(M_n(R)\), with only the \(j\)-th column being nonzero
Page: 245
Chapter: 7
Section: 3
Answer2: (a+b)*a in \(({a+b})^*\)
Index: 8
Kind: Ex

Question: \(A_1+A_2\) is
Answer: the smallest ideal containing the \(A_i\)'s
Page: 247
Chapter: 7
Section: 3
Variables: \(R\) a ring,
\(A_1,A_2\) ideals
Kind: Def

Smaller0: \(A_1/(A_1\cap A_2)\)
Greater0: \((A_1+ A_2)/A_1\)
Page: 245
Chapter: 7
Section: 3
Variables: \(R\) a ring,
\(A_1,A_2\) ideals
Index: 8
Kind: Theo
Smaller to greater: is isomorphic to
Name: Second isomorphism theorem

Smaller0: \((R/I)/(J/I)\)
Greater0: \(R/J\)
Page: 245
Chapter: 7
Section: 3
Variables: \(R\) a ring,
\(I\subseteq J\) ideals
Index: 8
Kind: Theo
Smaller to greater: is isomorphic to
Name: Third isomorphism theorem for rings

Question: \(J/I\) is
Answer: an ideal of \(R/I\)
Page: 245
Chapter: 7
Section: 3
Variables: \(R\) a ring,
\(I\subseteq J\) ideals
Index: 8
Kind: Theo

Smaller0: the set of rings \(A\) containing \(I\)
Greater0: the set of rings \(A/I\)
Page: 246
Chapter: 7
Section: 3
Variables: \(R\) a ring,
\(I\) an ideal
Index: 8 (3)
Kind: Theo
Smaller to greater: is isomorphic to
Name: Fourth isomorphism theorem for rings
Name2: Lattice isomorphism theorem for rings

Question: \(A_1A_2\) is
Answer: an ideal included in \(A_1\cap A_2\)
Page: 247
Chapter: 7
Section: 3
Variables: \(R\) a ring,
\(A_1,A_2\) ideals

Name: The ideal generated by \(A\)
Definition: The smallest ideal of \(R\) containing \(A\)
Page: 251
Chapter: 7
Section: 4
Variables: \(R\) a ring with identity, \(1\ne 0\),
\(A\subseteq R\)
Notation: \((A)\)
Index: 1
Kind: Def

Definition: The set of all finite sums of terms \(ra\) with \(r\in R\) and \(a\in A\)
Page: 251
Chapter: 7
Section: 4
Variables: \(R\) a ring with identity, \(1\ne 0\),
\(A\subseteq R\)
Notation: \(RA\)
Index: 2
Kind: Def

Definition: The set of all finite sums of terms \(rar'\) with \(r,r'\in R\) and \(a\in A\)
Page: 251
Chapter: 7
Section: 4
Variables: \(R\) a ring with identity, \(1\ne 0\),
\(A\subseteq R\)
Notation: \(RAR\)
Index: 2
Kind: Def

Name: A principal ideal
Definition: An ideal generated by a single element
Page: 251
Chapter: 7
Section: 4
Variables: \(R\) a ring with identity,
\(1\ne 0\),
Index: 3
Kind: Def
Example: 0
Example2: \(R\)

Name: A finitely generated ideal
Definition: An ideal generated by a finite set
Page: 251, 252
Chapter: 7
Section: 4
Variables: \(R\) a ring with identity
\(1\ne 0\)
Index: 3, 4
Kind: Def, Ex
Counterexample: In the ring of continuous functions, the ideal \(\{f\mid f(0)=0\}\)

Name: The ideal generated by \(A\)
Definition: \(RAR\)
Page: 251
Chapter: 7
Section: 4
Variables: \(R\) a ring,
\(A\subseteq R\)

Smaller0: \(R\) is commutative
Greater0: the set of \(R\) multiples of \(a\)
Page: 251
Chapter: 7
Section: 4
Variables: \(R\) a ring,
\(a\in R\)
Prefix Greater: the principal ideal generated by \(a\) is

Name: Ideal which are not principal
Page: 252
Chapter: 7
Section: 4
Index: 3
Kind: Exemple
Example: In \(\mathbb Z[X]\): \((2,x)\)
Example2: In \(\mathbb Z[X,Y]\): \((y,x)\)

Smaller0: \(I\) contains a unit
Greater0: \(I=R\)
Page: 253
Chapter: 7
Section: 4
Variables: \(R\) a ring
\(I\) an ideal of \(R\)
Index: 9
Kind: Prop
Smaller to greater: iff

Smaller0: 0 and \(R\)
Greater0: \(R\) is a field
Page: 253
Chapter: 7
Section: 4
Index: 9 (2)
Kind: Prop
Smaller to greater: iff
Prefix Smaller: \(R\)'s ideal are

Smaller0: \(R\) is a field
Greater0: injections or zero
Page: 253
Chapter: 7
Section: 4
Variables: \(R\) a ring
Index: 10
Kind: Cor
Smaller to greater: iff
Prefix Greater: \(R\)'s homomorphism are
Extra: definition

Smaller0: \(R\)'s left and right proper ideals are 0
Greater0: \(R\) is a division ring
Page: 253
Chapter: 7
Section: 4
Variables: \(R\) a ring with identity \(1\ne0\)

Name: A maximal ideal
Definition: A \(\subseteq\)-maximal proper ideal
Page: 253, 254, 255
Chapter: 7
Section: 4
Variables: \(R\) a ring
Index: , 12, 2, 4, 5
Kind: , example, prop, Example, example
Definition2: An ideal \(M\) such that \(R/M\) is a field (assuming \(R\) is commutative)
Example: \((2,x)\)
Example2: \(M_a=\{f\in R\to R'\mid f(a)=0\}\) for an arbitrary \(a\in R\)
Denoted by: \(\mathbb Z[x]\),
Applied to2: ring of function \(R\to R'\)

Question: A ring without maximal ideal
Answer: \((G,+,0)\) a group without maximal subgroup, using \(\times:(a,b)\mapsto 0\)
Page: 253
Chapter: 7
Section: 4

Smaller0: \(R\) has an identity
Greater0: \(I\) is contained in a maximal ideal
Page: 253
Chapter: 7
Section: 4
Variables: \(R\) a ring,
\(I\) a proper ideal
Index: 11
Kind: Prop
Assuming: Zorn's lemma
Extra: counter ?
Counterexample: \(2\mathbb Z\) as a ring, \(6\mathbb Z\) is a maximal ideal

Question: The principal ideals of \(\mathbb Z\) are
Answer: \(\mathbb Z/p\mathbb Z\) with \(p\) prime
Page: 255
Chapter: 7
Section: 4
Index: 1
Kind: Examples

Name: The kernel of evaluation at \(a\)
Definition: The set of \(f\in G^X\) such that \(f(a)=0\)
Page: 255
Chapter: 7
Section: 4
Variables: \((G,\cdot,0)\) a group,
\(X\) a set,
\(a\in X\)
Context: Ring theory
Notation: \(M_a\)
Kind: Examples

Smaller0: \(R\) is a field
Greater0: a maximal ideal of \(RG\)
Page: 255
Chapter: 7
Section: 4
Variables: \((G,\cdot,0)\) a group,
\(R\) a ring
Index: 5
Kind: Examples
Prefix Greater: the augmentation ideal of \(RG\) is
Context: Augmentation ideal:
\(\{\sum_{i=1}^na_ig_i\mid a_i\in R, \sum a_i=0_R\}\)

Name: A prime ideal
Definition: for each \(a,b\in R\), \(ab\in I\) implies either \(a\) or \(b\) belongs to \(I\).
Page: 255
Chapter: 7
Section: 4
Variables: \(R\) a ring
Index: , 13
Kind: Def, Prop
Definition2: \(R/I\) is an integral domain (assuming \(R\) commutative)
Conjdef: An ideal \(I\) such that
Assuming: \(R\) commutative

Smaller0: \(I\) is a maximal ideal
Greater0: \(I\) is a prime ideal
Page: 255, https://math.stackexchange.com/questions/146628/alternative-proof-of-i-maximal-implies-i-prime
Chapter: 7
Section: 4
Variables: \(R\) a ring,
\(I\) an ideal
Index: 13
Kind: Prop
Assuming: \(R\) commutative
Construction: Let \(ab \in I\).
Assume that \(a \not \in I\). Since \(I\) is maximal, \(I+(a)=R\). Thus there exists \(i \in I\) and \(r \in R\) such that \(i+ar=1\). Thus \(ib+abr=b\)
\(ib \in I\), \(abr \in I\). Thus its sum, \(b\), belongs to \(I\)
Counterexample: \((x)\) in \(\mathbb Z[x]\)

Question: Example of a prime non-maximal ideal of \(R\)
Answer: The ideal 0
Page: 255
Chapter: 7
Section: 4
Variables: \(R\) a ring
Index: 2
Kind: Example

Name: The radical of \(I\)
Definition: The set of elements \(r\) such that \(r^n\in i\) for some \(n\in\mathbb Z^+\)
Page: 258, https://en.wikipedia.org/wiki/Radical_of_an_ideal
Chapter: 7
Section: 4
Variables: \(R\) a ring,
\(I\) an ideal
Notation: \(\sqrt i\)
Index: 30
Kind: Exercice
Definition2: The pre-image of the ideal of nilpotent elements in \(R/I\)
Abbreviation: \(\text{rad }I\)
Intuition: The set of roots of elements of \(I\)
Typ: An ideal containing \(I\)
Assuming: \(R\) commutative

Name: An ideal radical
Definition: An ideal equal to its radical
Page: 258, https://en.wikipedia.org/wiki/Radical_of_an_ideal
Chapter: 7
Section: 4
Variables: \(R\) a ring
Index: 30
Kind: Exercice
Name2: A semiprime ideal
Extra: Even wp states it must be commutative
Assuming: \(R\) commutative

Definition: The intersection of all maximal ideals of \(R\) that contain \(I\)
Page: 259
Chapter: 7
Section: 4
Variables: \(R\) a ring,
\(I\) an ideal
Notation: Jac \(I\), \(J(I)\)
Index: 32
Kind: Exercice
Definition2: Preimage of the Jacobson radical of \(R/I\)
Extra: Jac for Jacobson, in case where \(I=0\)
Assuming: \(R\) commutative

Name: The left ideal generated by \(A\)
Definition: \(RA\)
Page: 251
Chapter: 7
Section: 4
Variables: \(R\) a ring,
\(A\subseteq R\)

Name: The right ideal generated by \(A\)
Definition: \(AR\)
Page: 251
Chapter: 7
Section: 4
Variables: \(R\) a ring,
\(A\subseteq R\)

Name: A division ring
Definition: \(\times\) admits an identity distinct from 0
Page: 224, 253
Chapter: 7, 7
Section: 1, 4
Kind: Definition
Definition2: every nonzero element has a multiplicative inverse
Conjdef: A ring \((R,+,\times)\) such that, both
Name2: A skew field
Typ: noncomutative field

Smaller0: \(X\) is a subring
Greater0: \(X\) is an integral domain
Page: 229
Chapter: 7
Section: 1
Variables: \(F\) a field and
\(X\subseteq F\).
Index: (5)
Kind: Ex
Smaller1: \(X\) contains the identity

Question: The degree of \(pq\) is
Answer: the sum of the degree of \(p\) and \(q\).
Page: 235
Chapter: 7
Section: 2
Variables: \(R\) a ring with unit
\(p,q\in R[x]\)
Index: 4
Kind: Prop
Assuming: \(R\) is an integral domain

Question: A basis of \(RG\)
Answer: \(G\)
Page: 841
Chapter: 18
Section: 1
Variables: \(R\) a commutative ring with identity \(1\ne 0\)
\(G\) a finite multiplicative group.

Name: The set of ideals
Page: 246, 247
Section: 3
Chapter: 7
Index: 8(1)
Variables: \(R\) a ring
Kind: Theo, Def
Under: intersection
Under2: product (e.g. finite sum of product of elements of the ideal)
Under3: sum
Under4: power by an integer

Name: The field of fractions of \(R\)
Definition: \(\left(R\setminus\{0\}\right)^{-1}R\)
Page: 263, 246
Chapter: 7
Section: 5
Variables: \(R\) a ring,
Index: 1, (2), (4)
Kind: Def, Ex
Example: \(\mathbb Q\)
Conjdef: :
Name2: The quotient field of \(R\)
Example2: the field of fraction is the quadratic field \(\mathbb Q(\sqrt D)\)
Assuming: \(R\) commutative integral domain
Example3: The set of rational functions in the variable \(x\) over \(R\)
DefType: The field
Applied to: \(\mathbb Z\)
Applied to3: Let \(R'\) an integral domain.
Let \(R=R'[x]\).
Applied to2: Let \(D\) square free,
For \(R\) the quadratic integer ring \(\mathcal O\)

Name: The rings of fractions of \(D\) with respect to \(R\)
Definition: \(R\subseteq Q\)
Page: 261, 263, 264
Chapter: 7
Section: 5
Variables: \(R\) a ring,
\(D\subseteq R\)
Notation: \(D^{-1}R\)
Index: 15, (3)
Kind: theo, def, Ex
Definition2: each element of \(D\) is unit in \(Q\)
Example: \(Q=\mathbb Q\)
Conjdef: all
Definition3: \(Q\) has an injective morphism to every other rings satisfying those properties
Intuition: The ring of elements of the form \(\frac rd\), with \(r,d\in R\), \(d\ne 0\). Addition and multiplication defined as usual for fractions.
Assuming: \(R\) commutative integral domain,
\(D\) closed under multiplication,
containing non zero divisors
DefType: A ring \(Q\)
Applied to: \(R=\mathbb Z\), \(D=2\mathbb Z\)

Name: The subfield generated by \(A\)
Definition: intersection of each subfields of \(F\) containing \(A\)
Page: 263
Chapter: 7
Section: 5
Variables: \(F\) a field,
\(A\subseteq F\)
Index: 1
Kind: Theo
Definition2: smallest containing \(A\)
Conjdef: equivalenty
DefType: The field:

Definition: \(R\left(\{d^i\mid i\in\mathbb N\}\right)^{-1}\)
Page: 263
Chapter: 7
Section: 5
Variables: \(R\) a ring,
\(d\in R\)
Notation: \(R[1/d]\)
Index: (5)
Kind: ex
Intuition: The set of elements of the form \(\frac{r}{d^i}\) with \(i\in\mathbb N\)
Assuming: \(R\) commutative, with a 1
\(d\) not a zero divisor

Page: 265
Chapter: 7
Section: 6
Name: The (ring) direct product
Set: \(\prod R_i\)
Variables: \(I\) a set,
\((R_i)_{i\in I}\) a family of rings
*: componentwise multiplication
Kind: ex
+: componentwise addition
0: Tuple of 0's
1: Tuple of 1's
Units: Tuple of units
Zero divisors: Either containing a 0, or tuple of 0 divisors

Name: Comaximal ideals
Definition: Ideals \(A,B\) such that \(A+B=R\)
Page: 265
Chapter: 7
Section: 6
Variables: \(R\) a ring
Kind: Def
Extra: Don't actually have to be maximal. Same than non-prime numbers can be co-prime
Assuming: \(R\) commutative

Question: \(\prod_{i\in I} (a_i)\) equals
Answer: \((\prod_{i\in I}a_i)\)
Page: 265
Chapter: 7
Section: 6
Variables: \(R\) a ring
\(I\) a family
\((a_i)_{i\in I}\in R\)
Kind: Def
Assuming: \(R\) commutative

Question: The kernel of [$]\defFun{R}{\prod_{i\in I} R/A_i}{r}{(r+A_i)_{i\in I}}[/$]
Answer: is \(\bigcap A_i\)
Page: 265
Chapter: 7
Section: 6
Variables: \(R\) a ring
\((A_i)_{i\in I}\) ideals
Index: 17
Kind: Theo

Smaller0: the ideals are pairwise comaximal
Greater0: [$]\defFun{R}{\prod_{i\in I} R/A_i}{r}{(r+A_i)_{i\in I}}[/$] is surjective
Page: 265
Chapter: 7
Section: 6
Variables: \(R\) a ring
\((A_i)_{i\in I}\) ideals
Index: 17
Kind: Theo
Name: Chinese Remainder Theorem
Greater1: \(\bigcap A_i=\prod A_i\)
Extra: \(I\) finite in the hypothesis, it seems useless in the proof
Greater2: \(R/(\prod A_i)\cong \prod (R/A_i)\)

Smaller0: \(\mathbb Z/(\prod p_i^{\alpha_i})\)
Greater0: \(\prod (\mathbb Z/p_i^{\alpha_i}\mathbb Z)\)
Page: 267
Chapter: 7
Section: 6
Variables: \((p_i)\) the sequence of primes
\((\alpha_i)\) a sequence of naturals with finite supports
Index: 18
Kind: Cor
Smaller to greater: \(\cong\)

Smaller0: \((\mathbb Z/(\prod p_i^{\alpha_i}))^\times\)
Greater0: \(\prod(\mathbb Z/p_i^{\alpha_i}\mathbb Z)^\times\)
Page: 267
Chapter: 7
Section: 6
Variables: \((p_i)\) the sequence of primes
\((\alpha_i)\) a sequence of naturals with finite supports
Index: 18
Kind: Cor
Smaller to greater: \(\cong\)

Smaller0: \(\phi(\prod p_i^{\alpha_i})\)
Greater0: \(\prod\phi(p_i^{\alpha_i})\)
Page: 267
Chapter: 7
Section: 6
Variables: \((p_i)\) the sequence of primes
\((\alpha_i)\) a sequence of naturals with finite supports
Index: 18
Kind: Cor
Smaller to greater: \(=\)

Name: An idempotent of \(R\)
Definition: \(e\in R\) such that \(e=e^2\)
Page: 267
Chapter: 7
Section: 6
Variables: \(R\) a ring
Index: 1
Kind: Ex

Name: A directed set
Definition: for each \(i,j\in X\), there is some \(k\in X\) with \(i,j\le k\)
Page: 268
Chapter: 7
Section: 6
Kind: Ex
Conjdef: An ordered set \(X\) such that

Name: A directed system
Definition: there exists \(\rho_{ij}:A_i\to A_j\)
Page: 268
Chapter: 7
Section: 6
Variables: \((I,\le)\) a directed set,
Index: 8 (a)
Kind: Ex
Definition2: \(\rho_{jk}\circ\rho_{ij}=\rho_{ik}\)
Conjdef: A set of \((A_i)_{i\in I}\) such that for each \(i\le j\le k\),
Definition3: \(\rho_{ii}\) is the identity

Definition: the relations on \(a_n\in A_{i_n}\), such that there are \(\rho_{i_nj}\) such that \(\rho_{i_1j}a_1=\rho_{i_2j}a_2\)
Page: 268
Chapter: 7
Section: 6
Variables: \((I,\le)\) a directed set,
\((A_i)_{i\in I}\) a directed system
Notation: \(\sim\)
Index: 8 (a)
Kind: Ex

Name: The direct limit of \((A_i)\)
Definition: Quotient of the disjoint union of the \(A_i\)'s under \((a_i,\rho_{i,j}(a_i))\)
Page: 268
Chapter: 7
Section: 6
Variables: \((I,\le)\) a directed set,
\((A_i)_{i\in I}, (\rho_{i,j}:A_i\to A_j)_{i\le i}\) a directed system
Notation: \(\underrightarrow{\lim}A_i\)
Index: 8 (a)
Kind: Ex
Name2: The inductive limit of \((A_i)\)

Name: The inverse limit of \((A_i)\)
Definition: The set of \(P\subseteq\prod _{i\in I}A_i\) of elements \((a_i)_{i\in I}\) such that \(\mu_{ji}(a_j)=a_i\)
Page: 269
Chapter: 7
Section: 6
Variables: \((I,\le)\) an ordered set,
\((A_i)_{i\in I}\) a directed system of groups
Context: Group theory
Notation: \(\underleftarrow{\lim}A_i\)
Index: 10
Kind: Ex
Example: the set of branches
Name2: The projective limit of \((A_i)\)
Typ: Subgroup of the direct product group
Applied to: directed tree

Name: A norm on the integral domain \(R\)
Definition: A function \(R\to\mathbb N\) sending 0 to 0
Page: 270
Chapter: 8
Section: 1
Variables: \(R\) a commutative integral domain
Context: integral domain
Extra: checked
Intuition: A size on it
Denoted by: \(N\)

Name: A positive norm on the integral domain \(R\)
Definition: A norm \(N:R\to\mathbb N\) such that \(N(x)=0\) iff \(x=0\)
Page: 270
Chapter: 8
Section: 1
Variables: \(R\) a commutative integral domain

Name: An Euclidean domain
Definition: \(R\) a commutative integral domain
Page: 270, 271, 273
Chapter: 8
Section: 1,
Index: 0, 1, 2, 3, 1
Kind: ex, ex, ex, ex, Prop
Definition2: \(N\) a norm such that for any \(a,b\in R\), \(b\ne 0\), there exists element \(q,r\in R\) with \(a=qb+r\) with \(r=0\) or \(N(r)<N(b)\)
Example: Fields, where \(q=ab^{-1}\)
Conjdef: A pair \((R,N)\) with
Name2: An integral domain possessing a division algorithm
Example2: \(\mathbb Z\) with [$]N(a)=\abs{a}[/$]
Example3: \(F[x]\) for \(F\) a field and \(N(p)\) the degree of \(p\)
DefType: The PID
Counterexample: \(R[x]\) for \(R\) a ring which is not a field
Example4: \(\mathbb Z[i]\) with \(N(a+bi)=a^2+b^2\)
Counterexample2: \(\mathscr O=\mathbb Z[\sqrt n]\) in general

Name: The remainder of \(a\) by \(b\)
Definition: A value \(r\in R\) such that there exists \(q\in R\) such that \(a=qb+r\), and \(r=0\) or \(N(r)<N(b)\)
Page: 270
Chapter: 8
Section: 1
Variables: \((R,N)\) an Euclidean domain
\(a,b\in R\)
Denoted by: \(r\)
Assuming: \(b\ne 0\)

Name: The quotient of \(a\) by \(b\)
Definition: A value \(q\in R\) such that there exists \(r\in R\) such that \(a=qb+r\), and \(r=0\) or \(N(r)<N(b)\)
Page: 270
Chapter: 8
Section: 1
Variables: \((R,N)\) an Euclidean domain
\(a,b\in R\)
Denoted by: \(q\)
Assuming: \(b\ne 0\)

Name: A discrete valuation on \(K\)
Definition: \(f(ab)=f(a)+f(b)\)
Page: 232, 272
Chapter: 7, 8
Section: 1, 1
Variables: \(K\) a field
Index: 26, 4
Kind: Ex, Ex
Definition2: \(f\) surjective
Conjdef: A function \(f:K^\times\to\mathbb Z\) satisfying
Definition3: \(f(x+y)\ge\min(f(x),f(y))\) for all \(x,y\in K^\times\) with \(x+y\ne0\)
Extra: Checked
Denoted by: \(\nu\)

Name: The valuation ring of \(\nu\)
Definition: The set containing \(0\) and the \(x\in K^\times\) such that \(\nu(x)\ge0\)
Page: 272
Chapter: 8
Section: 1
Variables: \(K\) a field
\(\nu\) a discrete valuation on \(K\)
Index: 4
Kind: Ex

Name: A discrete valuation ring
Definition: its field of fractions admits a valuation \(\nu\) such that
Page: 272
Chapter: 8
Section: 1
Index: 4
Kind: Ex
Definition2: \(R\) is the valuation ring of \(\nu\)
Example: The ring of rational numbers whose denominators are relatively prime to some fixed prime \(p\). Its field of fractions is \(\mathbb Q\) with \(\nu(\frac pq)=p\)
Conjdef: both
Typ: A Euclidean domain with norm \(N(0)=0\) and \(N=\nu\) otherwise
DefType: An integral domain \(R\)

Question: \(I\) is of the form
Answer: either \(\{0\}\), or of the form \((d)\) with \(d\) of norm minimal and positive
Page: 273
Chapter: 8
Section: 1
Variables: \((R,N)\) a Euclidean domain
\(I\subseteq R\) an ideal
Kind: Prop

Question: How to prove that \(\mathbb Z[x]\) is not an Euclidean domain for any norm
Answer: \((2,x)\) is not a principal ideal
Page: 273
Chapter: 8
Section: 1
Kind: Ex

Name: A multiple of \(a\)
Definition: \(ax\) for some \(x\in R\)
Page: 274
Chapter: 8
Section: 1
Variables: \(R\) a commutative ring
\(a\in R\)
Index: (1)
Kind: Def

Name: A divisor of \(a\)
Definition: An element \(b\) such that \(b=ax\)
Page: 274
Chapter: 8
Section: 1
Variables: \(R\) a commutative ring
\(a\in R\)
Index: (1)
Kind: Def

Name: \(a\) divides \(b\)
Definition: \(b=ax\) for some \(x\in R\)
Page: 274
Chapter: 8
Section: 1
Variables: \(R\) a ring
\(a,b\in R\)
Notation: \(a\mid b\)
Index: (1)
Kind: Def
Definition2: \((b)\subseteq(a)\)

Name: A least common multiple of \(a\) and \(b\)
Definition: \(e\mid a\), \(b\mid e\), and for each \(e'\) with \(a\mid e'\) and \(b\mid e'\), \(e\mid e'\)
Page: 279
Chapter: 8
Section: 1
Variables: \(R\) a ring
\(a,b\in R\)
Index: (2), 4
Kind: Def, Theo
Definition2: \((e)\) is the unique largest principal ideal contained in \((a)\cap (b)\)
Conjdef: A non zero \(e\in R\) such that equivalently
Definition3: \(\frac{ab}{(a,b)}\) assuming Euclidean domain
Abbreviation: \(\text{lcm}(a,b)\)

Name: A greatest common divisor of \(a\) and \(b\)
Definition: \(d\mid a\), \(d\mid b\), and for each \(d'\) with \(d'\mid a\) and \(d'\mid b\), \(d'\mid d\)
Page: 274, 275
Chapter: 8
Section: 1
Variables: \(R\) a ring
\(a,b\in R\)
Notation: \((a,b)\)
Index: (2), 4
Kind: Def, Theo
Definition2: the ideal \(I\) generated by \((a,b)\) is contained in \((d)\) and for any \(d'\) with \(I\subseteq(d')\), \((d)\subseteq(d')\)
Conjdef: A non zero \(d\ R\) such that equivalently
Definition3: \(r_n\) where
\(a=q_0b+r_0\),
\(r=q_1r_0+r_1\)
\(r_0=q_2r_1+r_2\), ....
\(r_{n-1}=q_{n+1}r_n\), assuming \(R\) an Euclidean domain
Abbreviation: \(\gcd(a,b)\)
Definition4: \((d)=(a,b)\) assuming Euclidean domain

Name: A Bezout domain
Definition: An integral domain in which every ideal \((a,b)\) is principal
Page: 274
Chapter: 8
Section: 1
Example: In \(\mathbb Z\), \((a\mathbb Z,b\mathbb Z)=(a,b)\mathbb Z\)

Smaller0: \((d)=(d')\)
Greater0: \(d'=ud\) for some unit \(u\)
Page: 274
Chapter: 8
Section: 1
Variables: \(R\) an integral domain,
\(d,d'\in R\)
Index: 3
Kind: Prop
Smaller to greater: iff

Smaller0: \(d\) and \(d'\) are both gcd of \(a\) and \(b\)
Greater0: \(d'=ud\) for some unit \(u\)
Page: 274
Chapter: 8
Section: 1
Variables: \(R\) an integral domain,
\(d,d'\in R\)
\(a,b\in R\)
Index: 3
Kind: Prop
Smaller to greater: iff

Definition: \(R^\times\cup\{0\}\)
Page: 277
Chapter: 8
Section: 1
Variables: \(R\) an integral domain,
Notation: \(\tilde R\)
Intuition: Elements of norm at most 1

Name: An universal side divisor
Definition: for every \(x\in R\) there is \(z\in\tilde R\) such that \(u\) divides \(x-z\) in \(R\). I.e. \(x=qu+z\) with \(z\) unit or zero
Page: 277
https://commalg.subwiki.org/wiki/Universal_side_divisor
Chapter: 8
Section: 1
Variables: \(R\) an integral domain,
Example: 2 and 3
Conjdef: Some \(u\in R\setminus\tilde R\) such that
Example2: \(ax\)
Denoted by: \(u\)
Applied to: \(\mathbb Z\)
Applied to2: \(F[X]\) for \(F\) a field

Smaller0: \(R\) is not a field
Greater0: at least 1
Page: 277
Chapter: 8
Section: 1
Variables: \(R\) an integral domain,
Index: 5
Kind: Prop
Prefix Greater: the number of universal side divisors in \(R\) is
Smaller1: \(R\) is an Euclidean domain
Construction: take element of \(R\setminus\tilde R\) of minimal norm

Smaller0: \(N(x)\) is minimal non-zero
Greater0: unit
Page: 278
Chapter: 8
Section: 1
Variables: \((R,N)\) an Euclidean domain
\(x\in R\)
Index: 3
Kind: Ex
Prefix Greater: \(x\) is
Smaller1: \(N(x)=0\)
Connect Smaller0: or
Construction: Assume \(N(x)=0\).
let \(q,r\) such that \(1=xq+r\) and \(N(r)<N(x)=0\) (impossible) or \(r=0\). Thus \(1\) is a multiple of \(x\).
Assume \(N(y)>0\), with \(y\) such that \(N(y)\) is minimal.\(1=yq+r\) with \(N(r)<N(x)\) or \(r=0\). In both case, \(N(r)=0\). Thus \(1=ra\). \(r(a-1)=yq\) ????

Smaller0: \((a,b)=1\)
Greater0: \(a\) divides \(c\)
Page: 278
Chapter: 8
Section: 1
Variables: \((R,N)\) an Euclidean domain
\(a,b,c\in R\)
Index: 4
Kind: Ex
Smaller1: \(a\mid bc\)

Smaller0: \(a,b\) are nonzero
Greater0: \(\frac{a}{(a,b)}\) divides \(c\)
Page: 278
Chapter: 8
Section: 1
Variables: \((R,N)\) an Euclidean domain
\(a,b,c\in R\)
Index: 4
Kind: Ex
Smaller1: \(a\mid bc\)

Smaller0: \(x_0,y_0\) is a solution of \(xa+by=N\)
Greater0: the set of \(\left(x_0+m\frac{b}{(a,b)},y_0-m\frac{a}{(a,b)}\right)\)
Page: 278
Chapter: 8
Section: 1
Variables: \(a,b,x_0,y_0,N\in\mathbb Z\),
\(a,b\ne 0\)
Index: 4
Kind: Ex
Smaller to greater: iff
Prefix Greater: the set of solutions of \(xa+by=N\) is

Smaller0: \(a,b\) relatively prime
Greater0: \(ab-a-b\)
Page: 278
Chapter: 8
Section: 1
Variables: \(a,b\in\mathbb N\)
Index: 6
Kind: Ex
Prefix Greater: the greatest number which is not of the form \(ax+by\) with \(x,y\in \mathbb N\) is
Name: The Postage Stamp Problem

Question: \(\mathbb Z[i]/I\) is
Answer: finite
Page: 278
Chapter: 8
Section: 1
Variables: \(I\) an ideal of \(\mathbb Z[i]\)
Index: 10
Kind: Ex
Assuming: \(I\) nonzero
Construction: \(I=(\alpha)\), a coset \(a+I\) is represented by an element of norm less than \(N(\alpha)\)

Page: 226
Chapter: 7
Section: 1
Name: The ring of integers
Notation: \(\mathbb Z\)
Units: \(\pm 1\)
Zero divisors: None

Page: 226
Chapter: 7
Section: 1
Name: The ring of integers modulo \(n\)
Notation: \(\mathbb Z/n\mathbb Z\)
Kind: Examples
Index: (2)
Units: Elements co-prime to \(n\)
Zero divisors: Divisors of \(n\) distinct from \(1\)

Page: 226
Chapter: 7
Section: 1
Name: Functions \([0,1]\to\mathbb R\)
Kind: Examples
Index: (2)
Units: Functions whose kernel is empty
Zero divisors: Functions whose kernel is non-empty

Page: 226
Chapter: 7
Section: 1
Name: Continuous functions \([0,1]\to\mathbb R\)
Kind: Examples
Index: (4)
Units: Functions whose kernel is empty
Zero divisors: Functions which are zero on an interval

Page: 226
Chapter: 7
Section: 1
Name: Quadratic field
Set: \(a+b\sqrt D\) with \(a,b\in\mathbb Q\)
Variables: \(D\) not a perfect square/square free
Notation: \(\mathbb Q(\sqrt D)\)
*: \((a+b\sqrt D)(a'+b'\sqrt D)\mapsto aa'+bb'D+(ab'+a'b)\sqrt D\)
Kind: Examples
Index: (5)
+: \((a+b\sqrt D)+(a'+b'\sqrt D)\mapsto (a+a')+(b+b')\sqrt D\)
0: 0
Units: Every non zero elements
÷: \(a+b\sqrt D\mapsto \frac{a-b\sqrt D}{a^2-Db^2}\)

Page: 229
Chapter: 7
Section: 1
Name: Quadratic integer rings
Set: \(a+b\frac{1+\sqrt D}2\) for \(a,b\in\mathbb Z\)
Variables: \(D\) square free
Notation: \(\mathbb Z\left[\frac{1+\sqrt D}{2}\right])\)
*: \((a+b\frac{1+\sqrt D}2)(a'+b'\frac{1+\sqrt D}2)\mapsto (aa'+bb'\frac{D-1}4)+(ab'+a'b+bb')\frac{1+\sqrt D}{2}\)
Kind: Examples
Index: (5)
+: \((a+b\frac{1+\sqrt D}2)+(a'+b'\frac{1+\sqrt D}2)\mapsto(a+a')+(b+b')\frac{1+\sqrt D}2\)
0: \(0+0\frac{1+\sqrt D}2\)
1: \(1+0\frac{1+\sqrt D}2\)
Units: Elements with norm 1
-: \(a+b\frac{1+\sqrt D}2\mapsto-a-b\frac{1+\sqrt D}2\)
Norm: \((a+b\frac{1+\sqrt D}2)\mapsto a^2-Db^2\)
Assuming: \(D\equiv 1\mod 4\)
Notation2: \(\mathcal O\)

Page: 229
Chapter: 7
Section: 1
Name: Quadratic integer rings
Variables: \(D\) square free
Notation: \(\mathbb Z\left[\sqrt D\right])\)
*: \((a+b\sqrt D)(a'+b'\sqrt D)\mapsto (aa'+bb'D)+(ab'+a'b)\sqrt D\)
Kind: Examples
Index: (5)
+: \((a+b\sqrt D)+(a'+b'\sqrt D)\mapsto (a+a')+(b+b')\sqrt D\)
0: \(0+0\sqrt B\)
1: \(1+0\sqrt B\)
Units: Elements with norm 1
-: \((a+b\sqrt D)\mapsto -a-b\sqrt D\)
Norm: \(a+b\sqrt D\mapsto a^2-Db^2\)
Assuming: \(D\equiv 2,3\mod 4\)

Question: Difference between \(\mathbb RQ_8\) and \(\mathbb H\)
Answer: In \(\mathbb H\), \(1_{\mathbb H}+ (-1)_{\mathbb H}=0_{\mathbb H}\) while in \(\mathbb RQ_8\), \(1_{\mathbb R}1_{Q_8}+1_{\mathbb R}(-1)_{Q_8}\) is distinct from \(0_{\mathbb RQ_8}\)
Page: 237
Chapter: 7
Section: 4
Kind: Example

Name: Universal side divisor
Page: https://commalg.subwiki.org/wiki/Universal_side_divisor
Under: taking an associate element