nyuuzyou/wikis huggingface dataset:
Properties of Norm on Division Ring/Norm of Negative of Unity

== Theorem ==
Let $\struct {R, +, \circ}$ be a division ring with unity $1_R$.
Let $\norm {\,\cdot\,}$ be a norm on $R$.
Then:
:$\norm {-1_R} = 1$
== Proof ==
By Product of Ring Negatives:
:$-1_R \circ -1_R = 1_R \circ 1_R = 1_R$
So:
Thus:
:$\norm {-1_R} = \pm 1$
By : 
:$\norm {-1_R} \ge 0$
Hence:
:$\norm {-1_R} = 1$
== Sources ==
* : $\S 2.2$: Basic Properties, Lemma $2.2.1\,iii)$
* : $\S 1.2$: Normed Fields, Theorem $1.6\,(a)$
Category:Properties of Norm on Division Ring
Properties of Norm on Division Ring/Norm of Power Equals Unity

== Theorem ==
Let $\struct {R, +, \circ}$ be a division ring with zero $0_R$ and unity $1_R$.
Let $\norm {\,\cdot\,}$ be a norm on $R$.
Let $x \in R$.
Then:
:$\forall n \in \N_{>0}: \norm {x^n} = 1 \implies \norm x = 1$
== Proof ==
Let $n \in \N_{>0}$.
Let $\norm {x^n} = 1$.
By :
:$\norm x^n = 1$
Since $\norm x \ge 0$, by Positive Real Complex Root of Unity:
:$\norm x = 1$
as desired.
== Sources ==
* : $\S 2.2$: Basic Properties, Lemma 2.2.1 ii)
Category:Properties of Norm on Division Ring
Properties of Norm on Division Ring/Norm of Quotient

== Theorem ==
Let $\struct {R, +, \circ}$ be a division ring with zero $0_R$ and unity $1_R$.
Let $\norm {\,\cdot\,}$ be a norm on $R$.
Let $x, y \in R$
Then:
:$y \ne 0_R \implies \norm {x y^{-1} } = \norm {y^{-1} x} = \dfrac {\norm x} {\norm y}$
== Proof ==
Let $y \ne 0_R$.
By then:
:$\norm y \ne 0$
So:
Similarly:
:$\norm {y^{-1} x} = \dfrac {\norm x} {\norm y}$
== Sources ==
* : $\S 1.2$: Normed Fields, Theorem $1.6 \ \text {(e)}$
Category:Properties of Norm on Division Ring
Properties of Norm on Division Ring/Norm of Unity

== Theorem ==
Let $\struct {R, +, \circ}$ be a division ring with unity $1_R$.
Let $\norm {\,\cdot\,}$ be a norm on $R$.
Then:
:$\norm {1_R} = 1$.
== Proof ==
By :
:$\forall x, y \in R: \norm {x \circ y} = \norm x \norm y$
In particular:
:$\norm {1_R} = \norm {1_R \circ 1_R} = \norm {1_R} \norm {1_R}$
By :
:$\norm {1_R} \ne 0$
So $\norm {1_R}$ has an inverse in $R$.
Multiplying by this inverse:
:$\norm {1_R} \norm {1_R} = \norm {1_R} \iff \norm {1_R} = 1$
as desired.
== Sources ==
* : $\S 2.2$: Basic Properties, Lemma $2.2.1 \ \text i)$
* : $\S 1.2$: Normed Fields, Theorem $1.6 \ \text {(a)}$
Category:Properties of Norm on Division Ring
Properties of Ordered Field

== Theorem ==
Let $\struct {F, +, \cdot}$ be an ordered field with unity $1$, zero $0$.
Denote the strict order by $<$ and the weak order by $\le$.
Let $\Char F$ denote the characteristic of $F$.
Then the following hold for all $x, y, z \in F$:
:$(1): \quad x < 0 \iff -x > 0$
:$(2): \quad x > y \iff x - y > 0$
:$(3): \quad x < y \iff -x > -y$
:$(4): \quad \paren {z < 0} \land \paren {x < y} \implies x z > y z$
:$(5): \quad x \ne 0 \implies x^2 > 0$
:$(6): \quad 1 > 0$
:$(7): \quad \Char F = 0$
:$(8): \quad x > y > 0 \iff y^{-1} > x^{-1} > 0$
== Proof ==
By definition of ordering, the relation $\le$ is:
:reflexive
:transitive
:antisymmetric
and furthermore, every pair of elements is comparable.
The order is compatible with $F$ in the sense that, for all $x, y, z, c \in F$:
:$x < y \implies x + z < y + z$
:$c > 0,\ x < y \implies c x < c y$
The proof is by repeated deduction from these properties.
* $(1): \quad x < 0 \iff -x > 0$:
Conversely:
* $(2): \quad x > y \iff x - y > 0$:
Conversely:
* $(3): \quad x < y \iff -x > -y$:
Conversely:
* $(4): \quad (z < 0) \land (x < y) \implies x z > y z$:
By parts 1 and 3 above, if $z < 0$, $x < y$ then $-z > 0$ and $-x > -y$.
Then:
:$x z = \paren {-x} \paren {-z} > \paren {-y} \paren {-z} = y z$
* $(5): \quad x \ne 0 \implies x^2 > 0$:
If $x > 0$, then:
:$x^2 = x \cdot x > x \cdot 0 = 0$
If $x < 0$, then by 1, $-x > 0$, so:
:$x^2 = \paren {-x} \cdot \paren {-x} > \paren {-x} \cdot 0 = 0$
* $(6): \quad 1 > 0$:
This is immediate from $(5)$, noting that $1 = 1^2$ is a square.
* $(7): \quad \Char F = 0$:
By $(6)$, we have:
:$0 < 1 < 1 + 1 < 1 + 1 + 1 < \cdots$
so $n \cdot 1 \ne 0$ for all $n \in \N$.
* $(8): \quad x > y > 0 \iff y^{-1} > x^{-1} > 0$:
First let $x > 0$.
 $x^{-1} < 0$.
Then by $(4)$:
:$0 = 0 \cdot x^{-1} > x \cdot x^{-1} = 1$
which contradicts $(6)$, so $x^{-1} > 0$.
Now let $x > y > 0$.
Then:
The converse follows upon interchanging $x^{-1} \leftrightarrow x$ and $y^{-1} \leftrightarrow y$.
Category:Ordered Fields
Properties of Ordered Group

== Theorem ==
Let $\struct {G, \circ, \preccurlyeq}$ be an ordered group with identity $e$.
Let $\prec$ be the reflexive reduction of $\preccurlyeq$.
The following properties hold:
=== Ordering Compatible with Group Operation is Strongly Compatible ===
==== Corollary ====
=== Inversion Mapping Reverses Ordering in Ordered Group ===
==== Corollary ====
=== Operating on Ordered Group Inequalities ===
=== Power Function Preserves Ordering in Ordered Group ===
== Also see ==
* Properties of Relation Compatible with Group Operation
* Properties of Ordered Ring
Category:Ordered Groups
Properties of Ordered Ring

== Theorem ==
Let $\struct {R, +, \circ, \le}$ be an ordered ring whose zero is $0_R$ and whose unity is $1_R$.
Let $U_R$ be the group of units of $R$.
Let $x, y, z \in \struct {R, +, \circ, \le}$.
Then the following properties hold:
:$(1): \quad x < y \iff x + z < y + z$. Hence $x \le y \iff x + z \le y + z$ (because $\struct {R, +, \le}$ is an ordered group).
:$(2): \quad x < y \iff 0 < y + \paren {-x}$. Hence $x \le y \iff 0 \le y + \paren {-x}$
:$(3): \quad 0 < x \iff \paren {-x} < 0$. Hence $0 \le x \iff \paren {-x} \le 0$
:$(4): \quad x < 0 \iff 0 < \paren {-x}$. Hence $x \le 0 \iff 0 \le \paren {-x}$
:$(5): \quad \forall n \in \Z_{>0}: x > 0 \implies n \cdot x > 0$
:$(6): \quad x \le y, 0 \le z: x \circ z \le y \circ z, z \circ x \le z \circ y$
:$(7): \quad x \le y, z \le 0: y \circ z \le x \circ z, z \circ y \le z \circ x$
=== Total Ordering ===
If, in addition, $\struct {R, +, \circ, \le}$ is totally ordered, the following properties also hold:
:$(8): \quad 0 < x \circ y \implies \paren {0 < x \land 0 < y} \lor \paren {x < 0 \land y < 0}$
:$(9): \quad x \circ y < 0 \implies \paren {0 < x \land y < 0} \lor \paren {x < 0 \land 0 < y}$
:$(10): \quad 0 \le x \circ x$. In particular, if $R$ is non-null and has a unity, $0_R < 1_R$
:$(11): \quad x \in U_R \implies 0 < x \iff 0 < x^{-1}, x \le 0 \iff x^{-1} \le 0$
== Proof ==
== Sources ==
* : Chapter $\text {IV}$: Rings and Fields: $23$. The Field of Rational Numbers: Theorem $23.11$
Category:Ordered Rings
Properties of Orthogonal Projection

== Theorem ==
=== Orthogonal Projection is Linear Transformation ===
=== Orthogonal Projection is Bounded ===
=== Fixed Points of Orthogonal Projection ===
=== Orthogonal Projection is Projection ===
=== Kernel of Orthogonal Projection ===
=== Range of Orthogonal Projection ===
=== Orthogonal Projection onto Orthocomplement ===
== Sources ==
* : $\text{I}$ Hilbert Spaces: $\S 2.$ Orthogonality: Theorem $2.7$
Category:Orthogonal Projections
Properties of Prime Subfield

== Theorem ==
Let $F$ be a field.
Let $K$ be the prime subfield of $F$.
Then $K$ is isomorphic to either:
:$\Q$, the field of rational numbers, or
:$\Z_p$, the Ring of Integers Modulo $p$, where $p$ is prime.
== Proof ==
From Field of Characteristic Zero has Unique Prime Subfield, if $\Char F = 0$, then its prime subfield is isomorphic to $\Q$, the field of rational numbers.
From Field of Prime Characteristic has Unique Prime Subfield, if $\Char F = p$, then its prime subfield is isomorphic to $\Z_p$, the Ring of Integers Modulo $p$.
From Characteristic of Field is Zero or Prime, $p$ is prime.
== Sources ==
* : $\S 61.3$ Characteristic of an integral domain or field
Category:Subfields
Properties of Probability Generating Function

== Theorem ==
Let $X$ be a discrete random variable whose probability generating function is $\Pi_X \left({s}\right)$.
Then $\Pi_X \left({s}\right)$ has the following properties:
=== Probability Generating Function defines Probability Distribution ===
=== PGF of 0 ===
=== PGF of 1 ===
Category:Probability Generating Functions
Properties of Product of Identity plus Operator Raised to Powers of 2

== Theorem ==
Let $X$ be a Banach space.
Let $\map \LL X$ be the set of all linear transformations.
Let $\map {CL} X$ be a continuous linear transformation sapce.
Let $\norm {\, \cdot \,}$ be the supremum operator norm.
Let $A \in \map {CL} X$ be such that $\norm A < 1$.
Let $I$ be the identity mapping.
Let $\circ$ be the composition of mappings.
For all $n \in \N$ let $A^n := \underbrace {A \circ A \circ \ldots \circ A}_{n \text{ times}}$, with $A^0 := I$.
For all $n \in \N$ let $P_n = \underbrace{\paren {I + A} \circ \paren {I + A^2} \circ \ldots \circ \paren {I + A^{2^n} } }_{n + 1 \text{ terms} }$
Then:
:$\forall n \in \N : \paren {I - A} \circ P_n = I - A^{2^{n + 1} }$
:the sequence $\sequence {P_n}_{n \mathop \in \N}$ converges in $\map {CL} X$ to $\paren {I - A}^{-1}$
== Proof ==
=== $\paren {I - A} \circ P_n = I - A^{2^{n + 1} }$ ===
This will be a proof by induction.
==== Basis for the induction ====
Let $n = 0$.
Then:
==== Induction hypothesis ====
The induction hypothesis is the following statement:
:$\paren {I - A} \circ P_n = I - A^{2^{n + 1} }$
from which it is to be shown that:
:$\paren {I - A} \circ P_{n + 1} = I - A^{2^{\paren {n + 1} + 1} }$
==== Induction Step ====
=== $\sequence {P_n}_{n \mathop \in \N}$ converges to $\paren {I - A}^{-1}$ ===
We have that $\norm A < 1$.
Furthermore:
We have that Set of Linear Transformations with Supremum Operator Norm is Normed Vector Space.
Hence, $\sequence {I - A^{2^{n + 1} } }_{n \mathop \in \N}$ converges in $\struct {\map \LL X, \norm {\, \cdot \,}}$ to $I$.
By Neumann Series Theorem, $I - A$ is invertible in $\map {CL} X$.
Moreover:
Since:
Thus, $\sequence {P_n}_{n \mathop \in \N}$ converges to $\paren {I - A}^{-1}$ in $\struct {\map {CL} X, \norm {\, \cdot \,} }$.
== Sources ==
* : Chapter $\S 2.4$: Composition of continuous linear transformations
Category:Continuous Linear Transformations
Category:Banach Spaces
Category:Proofs by Induction
Category:Convergent Sequences
Category:Inverse Mappings
Properties of Rational Numbers

== Rational Numbers are Countably Infinite ==
== Rational Numbers under Addition form Infinite Abelian Group ==
=== Rational Addition is Closed ===
=== Rational Addition is Associative ===
=== Rational Addition Identity is Zero ===
=== Inverse for Rational Addition ===
=== Rational Addition is Commutative ===
== Non-Zero Rational Numbers under Multiplication form Infinite Abelian Group ==
=== Rational Multiplication is Closed ===
=== Rational Multiplication is Associative ===
=== Rational Multiplication Identity is One ===
=== Inverse for Rational Multiplication ===
=== Rational Multiplication is Commutative ===
== Strictly Positive Rational Numbers under Multiplication form Countably Infinite Abelian Group ==
== Rational Numbers form Ring ==
== Rational Numbers form Integral Domain ==
=== Rational Multiplication Distributes over Addition ===
== Rational Numbers form Ordered Integral Domain ==
== Rational Numbers form Totally Ordered Field ==
== Substructures and Superstructures ==
=== Additive Group of Integers is Normal Subgroup of Rationals ===
=== Additive Group of Rationals is Normal Subgroup of Reals ===
=== Additive Group of Rationals is Normal Subgroup of Complex ===
=== Multiplicative Group of Rationals is Normal Subgroup of Reals ===
=== Multiplicative Group of Rationals is Normal Subgroup of Complex ===
=== Integers form Subdomain of Rationals ===
=== Rational Numbers form Subfield of Real Numbers ===
== Also see ==
* Properties of Natural Numbers
* Properties of Integers
* Properties of Real Numbers
* Properties of Complex Numbers
* Properties of Hamiltonian Numbers
Category:Rational Numbers
Properties of Real Cosine Function

== Theorem ==
Let $x \in \R$ be a real number.
Let $\cos x$ be the cosine of $x$.
Then:
=== Cosine Function is Continuous ===
=== Cosine Function is Absolutely Convergent ===
=== Cosine of Zero is One ===
=== Cosine Function is Even ===
=== Cosine of Multiple of Pi ===
=== Cosine of Half-Integer Multiple of Pi ===
=== Shape of Cosine Function ===
== Also see ==
* Properties of Real Sine Function
Category:Cosine Function
Properties of Real Numbers

== Real Numbers are Uncountable ==
== Real Numbers under Addition form Infinite Abelian Group ==
=== Real Addition is Well-Defined ===
=== Real Addition is Closed ===
=== Real Addition is Associative ===
=== Real Addition Identity is Zero ===
=== Inverse for Real Addition ===
=== Real Addition is Commutative ===
== Non-Zero Real Numbers under Multiplication form Abelian Group ==
=== Real Multiplication is Well-Defined ===
=== Real Multiplication is Closed ===
=== Real Multiplication is Associative ===
=== Real Multiplication Identity is One ===
=== Inverse for Real Multiplication ===
=== Real Multiplication is Commutative ===
== Strictly Positive Real Numbers under Multiplication form Uncountable Abelian Group ==
== Real Numbers form Ring ==
=== Real Multiplication Distributes over Addition ===
== Real Numbers form Integral Domain ==
== Real Numbers form Ordered Integral Domain ==
== Real Numbers form Totally Ordered Field ==
== Substructures and Superstructures ==
=== Additive Group of Integers is Normal Subgroup of Reals ===
=== Additive Group of Rationals is Normal Subgroup of Reals ===
=== Additive Group of Reals is Normal Subgroup of Complex ===
=== Multiplicative Group of Rationals is Normal Subgroup of Reals ===
=== Multiplicative Group of Reals is Normal Subgroup of Complex ===
=== Rational Numbers form Subfield of Real Numbers ===
=== Real Numbers form Subfield of Complex Numbers ===
== Further Structural Properties ==
=== Real Numbers form Vector Space ===
=== Real Numbers form Algebra ===
== Also see ==
* Properties of Natural Numbers
* Properties of Integers
* Properties of Rational Numbers
* Properties of Complex Numbers
* Properties of Hamiltonian Numbers
Category:Real Numbers
Properties of Real Sine Function

== Theorem ==
Let $\sin: \R \to \R$ denote the real sine function.
Then:
=== Real Sine Function is Continuous ===
=== Sine Function is Absolutely Convergent ===
=== Sine of Zero is Zero ===
=== Sine Function is Odd ===
=== Sine of Multiple of Pi ===
=== Sine of Half-Integer Multiple of Pi ===
=== Shape of Sine Function ===
== Also see ==
* Properties of Real Cosine Function
Category:Sine Function
Properties of Relation Compatible with Group Operation

== Theorem ==
Let $\struct {G, \circ}$ be a group with identity element $e$.
Let $\RR$ be a relation on $G$ which is compatible with $\circ$.
The following properties hold:
=== Relation Compatible with Group Operation is Strongly Compatible ===
==== Corollary ====
=== Inverses of Elements Related by Compatible Relation ===
==== Corollary ====
Category:Compatible Relations
Category:Group Theory
Category:Relations Compatible with Group Operation
Properties of Relation Not Preserved by Restriction

== Theorem ==
If a relation is:
* serial,
* non-reflexive,
* non-symmetric,
* non-transitive or
* non-connected
it is impossible to state without further information whether or not any restriction of that relation has the same properties.
== Proof ==
=== Restriction of Serial Relation is Not Necessarily Serial ===
=== Restriction of Non-Reflexive Relation is Not Necessarily Non-Reflexive ===
=== Restriction of Non-Symmetric Relation is Not Necessarily Non-Symmetric ===
=== Restriction of Non-Transitive Relation is Not Necessarily Non-Transitive ===
=== Restriction of Non-Connected Relation is Not Necessarily Non-Connected ===
== Also see ==
* Properties of Restriction of Relation for properties which are preserved by restriction.
Category:Restrictions
Properties of Restriction of Relation

== Theorem ==
Let $S$ be a set.
Let $\RR \subseteq S \times S$ be a relation on $S$.
Let $T \subseteq S$ be a subset of $S$.
Let $\RR {\restriction_T} \subseteq T \times T$ be the restriction of $\RR$ to $T$.
If $\RR$ on $S$ has any of the properties:
* Reflexive
* Antireflexive
* Symmetric
* Antisymmetric
* Asymmetric
* Transitive
* Antitransitive
* Connected
... then $\RR {\restriction_T}$ on $T$ has the same properties.
== Proof ==
=== Reflexivity ===
==== Restriction of Reflexive Relation is Reflexive ====
==== Restriction of Antireflexive Relation is Antireflexive ====
=== Symmetry ===
==== Restriction of Symmetric Relation is Symmetric ====
==== Restriction of Asymmetric Relation is Asymmetric ====
==== Restriction of Antisymmetric Relation is Antisymmetric ====
=== Transitivity ===
==== Restriction of Transitive Relation is Transitive ====
==== Restriction of Antitransitive Relation is Antitransitive ====
=== Connectedness ===
==== Restriction of Connected Relation is Connected ====
== Also see ==
=== Properties of Relation Not Preserved by Restriction ===
Category:Restrictions
Properties of Riemann Surface

== Theorem ==
This page gathers some results about Riemann surfaces.
=== Riemann Surface is Path-Connected ===
=== Radó's Theorem ===
=== Conformal Isomorphism of Universal Cover of Riemann Surface ===
=== Riemann Surface is Metrizable ===
=== Riemann Surface admits Metric of Constant Curvature ===
Category:Riemann Surfaces
Properties of Rising Factorial

== Theorem ==
Let $x^{\overline n}$ denote the $n$th rising factorial power of $x$.
This page gathers together some of the properties of the rising factorial.
=== Rising Factorial in terms of Falling Factorial ===
=== Rising Factorial in terms of Falling Factorial of Negative ===
=== Rising Factorial as Quotient of Factorials ===
=== One to Integer Rising is Integer Factorial ===
=== Number to Power of One Rising is Itself ===
=== Number to Power of Zero Rising is One ===
=== Sum of Indices of Rising Factorial ===
== Also see ==
* Properties of Falling Factorial
Category:Rising Factorials
Properties of Semi-Inner Product

== Theorem ==
Let $V$ be a vector space over $\Bbb F \in \set {\R, \C}$.
Let $\innerprod \cdot \cdot$ be a semi-inner product on $V$.
Denote, for $x \in V$, $\norm x := \innerprod x x^{1 / 2}$.
Then, $\forall x, y \in V, a \in \Bbb F$:
:$(1): \quad \norm {x + y} \le \norm x + \norm y$
:$(2): \quad \norm {a x} = \size a \norm x$
== Proof ==
=== Proof of $(1)$ ===
For $x, y \in V$, compute:
Taking square roots on either side gives the result.
=== Proof of $(2)$ ===
For $x \in V$, $a \in \Bbb F$, compute:
Taking square roots on either side gives the result.
== Sources ==
* : $\text{I}$ Hilbert Spaces: $\S 1.$ Elementary Properties and Examples: Corollary $1.5$
Category:Semi-Inner Product Spaces
Properties of Strict Negativity

== Theorem ==
Let $\struct {D, +, \times}$ be an ordered integral domain, whose (strict) positivity property is denoted $P$.
Let $\le$ be the total ordering induced by $P$, and let $<$ be its strict total ordering counterpart.
Let $N$ be the (strict) negativity property on $D$:
:$\forall a \in D: \map N a \iff \map P {-a}$
Then the following properties apply for all $a, b \in D$:
 
=== Strict Negativity is equivalent to Strictly Preceding Zero ===
=== Strict Negativity is equivalent to Strict Positivity of Negative ===
=== Sum of Strictly Negative Elements is Strictly Negative ===
=== Product of Two Strictly Negative Elements is Strictly Positive ===
=== Product of Strictly Negative Element with Strictly Positive Element is Strictly Negative ===
== Sources ==
* : $\S 2.7$: Theorem $10$
Category:Ordered Integral Domains
Properties of Well Inside Relation

== Theorem ==
Let $L = \struct{S, \vee, \wedge, \preceq}$ be a distributive lattice with greatest element $\top$ and smallest element $\bot$.
Let $\eqslantless$ denote the well inside relation on $L$.
Then the following hold:
=== Element Well Inside Itself Iff Has Complement ===
=== Well Inside Implies Predecessor ===
=== Well Inside Relation Extends to Predecessor and Successor ===
=== Well Inside Elements Form Filter ===
=== Elements Well Inside Form Ideal ===
== Sources ==
* : Chapter $\text {III}$: Compact Hausdorff Spaces, $\S1.1$
Category:Well Inside Relation
Property of 490,689

== Theorem ==
The number $490 \, 689$ can be expressed as the sum of $3$ cubes in $2$ different ways:
:$490 \, 689 = 4^3 + 60^3 + 65^3 = 8^3 + 25^3 \times 78^3$
while at the same time the products of the contributory cube roots of each of those $2$ ways are equal:
:$4 \times 60 \times 65 = 8 \times 25 \times 78$
== Proof ==
Then:
== Historical Note ==
== Sources ==
* : $490,689$
Category:Cube Numbers
Category:490,689
Property of Being an Ideal is not Transitive

== Theorem ==
Let $J_1$ be an ideal of a ring $R$.
Let $J_2$ be an ideal of $J_1$.
Then $J_2$ need not necessarily be an ideal of $R$.
== Proof ==
Let $R = \Q \sqbrk X$ be the ring of polynomials in one variable $X$ over $\Q$.
Let:
:$J_1 = \set {a_0 + a_1 X + \cdots + a_n X^n \in R : a_0 = a_1 = 0}$
and
:$J_2 = \set {a_0 + a_1 X + \cdots + a_n X^n \in R : a_0 = a_1 = a_3 = 0}$
First let us show that $J_1$ is an ideal of $R$.
We establish the properties of the ideal test in order.
$(1): \quad J_1 \ne \O$
This follows from the fact that $X^2 \in J_1$.
$(2): \quad \forall P, Q \in J_1: P + \paren {-Q} \in J_1$
Let:
:$\ds P = \sum_{i \mathop = 0}^{+\infty} a_i X^i \in J_1$
:$\ds Q = \sum_{i \mathop = 0}^{+\infty} b_i X^i \in J_1$
Then by the definition of addition of polynomials:
:$\ds P + \paren {-Q} = \sum_{i \mathop = 0}^{+\infty} c_i X^i$
where:
:$c_i = a_i - b_i$
By assumption, $a_0 = a_1 = b_0 = b_1 = 0$.
Therefore:
:$c_0 = a_0 - b_0 = 0$
:$c_1 = a_1 - b_1 = 0$
Therefore $P + \paren {-Q} \in J_1$.
$(3): \quad \forall P \in J_1, Q \in R: Q \cdot P \in J_1$
Let
:$\ds P = \sum_{i \mathop = 0}^{+\infty} a_i X^i \in J_1$
:$\ds Q = \sum_{i \mathop = 0}^{+\infty} b_i X^i \in R$
By the definition of multiplication of polynomials:
:$\ds Q \cdot P = \sum_{i \mathop = 0}^{+\infty} c_i X^i$
where:
:$c_i = \sum_{j + k \mathop = i} a_j b_k$
In particular, since $a_0 = 0$:
:$c_0 = a_0 b_0 = 0$
Since $a_0 = a_1 = 0$:
:$c_ 1 = a_0 b_1 + a_1 b_0 = 0$
Therefore $Q \cdot P \in J_1$.
This shows that $J_1$ is an ideal of $R$.
Next we show that $J$ is an ideal of $J_1$.
Again, we verify the properties of the ideal test in turn.
$(1): \quad J_2 \ne \O$
This follows from the fact that $X^2 \in J_2$.
$(2): \quad \forall P, Q \in J_2: P + \paren {-Q} \in J_2$
Let:
:$\ds P = \sum_{i \mathop = 0}^{+\infty} a_i X^i \in J_2$
:$\ds Q = \sum_{i \mathop = 0}^{+\infty} b_i X^i \in J_2$
Then by the definition of addition of polynomials:
:$\ds P + \paren {-Q} = \sum_{i \mathop = 0}^{+\infty} c_i X^i$
where:
:$c_i = a_i - b_i$
By assumption:
:$a_0 = a_1 = a_3 = b_0 = b_1 = b_3 = 0$
Therefore:
:$c_0 = a_0 - b_0 = 0$
:$c_1 = a_1 - b_1 = 0$
:$c_3 = a_3 - b_3 = 0$
Therefore $P + \paren {-Q} \in J_2$.
$(3): \quad \forall P \in J_2, Q \in J_1: Q \cdot P \in J_2$
Let
:$\ds P = \sum_{i \mathop = 0}^{+\infty} a_i X^i \in J_2$
:$\ds Q = \sum_{i \mathop = 0}^{+\infty} b_i X^i \in J_1$
By the definition of multiplication of polynomials,
:$\ds Q \cdot P = \sum_{i \mathop = 0}^{+\infty} c_i X^i$
where:
:$c_i = \sum_{j + k \mathop = i} a_j b_k$
Since $J_2 \subseteq J_1$, we have $P, Q \in J_1$.
So we have already that $c_0 = c_1 = 0$.
Moreover $a_0 = a_1 = b_0 = b_1 = 0$, so
:$c_3 = a_0b_3 + a_1 b_2 + a_2 b_1 + a_3 b_0 = 0$
Therefore $Q \cdot P \in J_2$.
This shows that $J_2$ is an ideal of $J_1$.
Finally we wish to see that $J_2$ is not an ideal of $R$.
We have that $X^2 \in J_2$ and $X \in R$.
 $J_2$ were an ideal of $R$.
This would imply that:
:$X \cdot X^2 = X^3 \in J_2$
But by the definition of $J_2$, we must have that the coefficient of $X^3$ is $0$.
This is a contradiction, so $J_2$ is not an ideal of $R$.
Category:Ideal Theory
Property of Group Automorphism which Fixes Identity Only

== Theorem ==
Let $G$ be a finite group whose identity is $e$.
Let $\phi: G \to G$ be a group automorphism.
Let $\phi$ have the property that:
:$\forall g \in G \setminus \set e: \map \phi t \ne t$
That is, the only fixed element of $\phi$ is $e$.
Then:
:$\forall x, y \in G: x^{-1} \, \map \phi x = y^{-1} \, \map \phi y \implies x = y$
=== Corollary $1$ ===
=== Corollary $2$ ===
=== Corollary $3$ ===
== Proof ==
== Sources ==
* : Chapter $8$: Homomorphisms, Normal Subgroups and Quotient Groups: Exercise $26$
Category:Group Automorphisms
Category:Property of Group Automorphism which Fixes Identity Only