nyuuzyou/wikis huggingface dataset:
Primitives involving a x squared plus b x plus c

== Theorem ==
This page gathers together the primitives of some expressions involving $a x^2 + b x + c$.
=== Primitive of Reciprocal of $a x^2 + b x + c$ ===
=== Primitive of $x$ over $a x^2 + b x + c$ ===
=== Primitive of $x^2$ over $a x^2 + b x + c$ ===
=== Primitive of $x^m$ over $a x^2 + b x + c$ ===
=== Primitive of Reciprocal of $x \left({a x^2 + b x + c}\right)$ ===
=== Primitive of Reciprocal of $x^2 \left({a x^2 + b x + c}\right)$ ===
=== Primitive of Reciprocal of $x^n \left({a x^2 + b x + c}\right)$ ===
=== Primitive of Reciprocal of $\left({a x^2 + b x + c}\right)^2$ ===
=== Primitive of $x$ over $\left({a x^2 + b x + c}\right)^2$ ===
=== Primitive of $x^2$ over $\left({a x^2 + b x + c}\right)^2$ ===
=== Primitive of $x^m$ over $\left({a x^2 + b x + c}\right)^n$ ===
=== Primitive of $x^{2 n - 1}$ over $\left({a x^2 + b x + c}\right)^n$ ===
=== Primitive of Reciprocal of $x \left({a x^2 + b x + c}\right)^2$ ===
=== Primitive of Reciprocal of $x^2 \left({a x^2 + b x + c}\right)^2$ ===
=== Primitive of Reciprocal of $x^m \left({a x^2 + b x + c}\right)^n$ ===
== Also see ==
* Primitives involving $\sqrt{a x^2 + b x + c}$ 
Category:Primitives
Category:Primitives of Rational Functions
Category:Primitives involving a x squared plus b x plus c‎
Primitives involving x squared minus a squared

== Theorem ==
This page gathers together the primitives of some expressions involving $x^2 - a^2$.
=== Primitive of Reciprocal of $x^2 - a^2$: $\coth^{-1}$ form ===
=== Primitive of Reciprocal of $x^2 - a^2$: Logarithm Form ===
=== Primitive $x$ over $x^2 - a^2$ ===
=== Primitive $x^2$ over $x^2 - a^2$ ===
=== Primitive $x^3$ over $x^2 - a^2$ ===
=== Primitive of Reciprocal of $x \paren {x^2 - a^2}$ ===
=== Primitive of Reciprocal of $x^2 \paren {x^2 - a^2}$ ===
=== Primitive of Reciprocal of $x^3 \paren {x^2 - a^2}$ ===
== Also see ==
* Primitives involving $\paren {x^2 - a^2}^2$ 
* Primitives involving $\paren {x^2 - a^2}^n$ 
Category:Primitives
Category:Primitives of Rational Functions
Category:Primitives involving x squared minus a squared‎
Primitives involving x squared minus a squared squared

== Theorem ==
This page gathers together the primitives of some expressions involving $\paren {x^2 - a^2}^2$.
=== Primitive of Reciprocal of $\paren {x^2 - a^2}^2$ ===
=== Primitive $x$ over $\paren {x^2 - a^2}^2$ ===
=== Primitive $x^2$ over $\paren {x^2 - a^2}^2$ ===
=== Primitive $x^3$ over $\paren {x^2 - a^2}^2$ ===
=== Primitive of Reciprocal of $x \paren {x^2 - a^2}^2$ ===
=== Primitive of Reciprocal of $x^2 \paren {x^2 - a^2}^2$ ===
=== Primitive of Reciprocal of $x^3 \paren {x^2 - a^2}^2$ ===
== Also see ==
* Primitives involving $x^2 - a^2$
* Primitives involving $\paren {x^2 - a^2}^n$ 
Category:Primitives
Category:Primitives of Rational Functions
Category:Primitives involving x squared minus a squared‎
Primitives involving x squared plus a squared

== Theorem ==
This page gathers together the primitives of some expressions involving $x^2 + a^2$.
=== Primitive of Reciprocal of $x^2 + a^2$: $\arctan$ Form ===
=== Primitive of Reciprocal of $x^2 + a^2$: $\arccot$ Form ===
=== Primitive $x$ over $x^2 + a^2$ ===
=== Primitive $x^2$ over $x^2 + a^2$ ===
=== Primitive $x^3$ over $x^2 + a^2$ ===
=== Primitive of Reciprocal of $x \paren {x^2 + a^2}$ ===
=== Primitive of Reciprocal of $x^2 \paren {x^2 + a^2}$ ===
=== Primitive of Reciprocal of $x^3 \paren {x^2 + a^2}$ ===
== Also see ==
* Primitives involving $\paren {x^2 + a^2}^2$ 
* Primitives involving $\paren {x^2 + a^2}^n$ 
Category:Primitives
Category:Primitives of Rational Functions
Category:Primitives involving x squared plus a squared‎
Primitives involving x squared plus a squared squared

== Theorem ==
This page gathers together the primitives of some expressions involving $\left({x^2 + a^2}\right)^2$.
=== Primitive of Reciprocal of $\left({x^2 + a^2}\right)^2$ ===
=== Primitive $x$ over $\left({x^2 + a^2}\right)^2$ ===
=== Primitive $x^2$ over $\left({x^2 + a^2}\right)^2$ ===
=== Primitive $x^3$ over $\left({x^2 + a^2}\right)^2$ ===
=== Primitive of Reciprocal of $x \left({x^2 + a^2}\right)^2$ ===
=== Primitive of Reciprocal of $x^2 \left({x^2 + a^2}\right)^2$ ===
=== Primitive of Reciprocal of $x^3 \left({x^2 + a^2}\right)^2$ ===
== Also see ==
* Primitives involving $x^2 + a^2$ 
* Primitives involving $\left({x^2 + a^2}\right)^n$ 
Category:Primitives
Category:Primitives of Rational Functions
Category:Primitives involving x squared plus a squared‎
Primitives of Functions involving Power of Root of a x + b

== Theorem ==
This page gathers together the primitives of some rational functions involving $\sqrt {a x + b}$.
=== Primitive of $\left({\sqrt {a x + b} }\right)^m$ ===
=== Primitive of $x \left({\sqrt {a x + b} }\right)^m$ ===
=== Primitive of $x^2 \left({\sqrt {a x + b} }\right)^m$ ===
=== Primitive of $\dfrac {\left({\sqrt {a x + b} }\right)^m} x$ ===
=== Primitive of $\dfrac {\left({\sqrt {a x + b} }\right)^m} {x^2}$ ===
=== Primitive of $\dfrac {\left({\sqrt {a x + b} }\right)^m} {x^2}$ ===
=== Primitive of $\dfrac 1 {x \left({\sqrt {a x + b} }\right)^m}$ ===
== Also see ==
* Primitives of Functions involving $\sqrt {a x + b}$
Category:Primitives‎
Category:Primitives involving Root of a x + b‎
Primitives of Functions involving Root of a x + b

== Theorem ==
This page gathers together the primitives of some rational functions involving $\sqrt {a x + b}$.
=== Primitive of $\dfrac 1 {\sqrt {a x + b} }$ ===
=== Primitive of $\dfrac x {\sqrt {a x + b} }$ ===
=== Primitive of $\dfrac {x^2} {\sqrt {a x + b} }$ ===
=== Primitive of $\dfrac 1 {x \sqrt {a x + b} }$ ===
=== Primitive of $\dfrac 1 {x^2 \sqrt {a x + b} }$ ===
=== Primitive of $\sqrt {a x + b}$ ===
=== Primitive of $x \sqrt {a x + b}$ ===
=== Primitive of $x^2 \sqrt {a x + b}$ ===
=== Primitive of $\dfrac {\sqrt {a x + b} } x$ ===
=== Primitive of $\dfrac {\sqrt {a x + b} } {x^2}$ ===
=== Primitive of $\dfrac {x^m} {\sqrt {a x + b} }$ ===
=== Primitive of $\dfrac 1 {x^m \sqrt {a x + b} }$ ===
=== Primitive of $x^m \sqrt {a x + b}$ ===
== Primitive of $\dfrac {\sqrt {a x + b} } {x^m}$ ==
== Also see ==
* Primitives of Functions involving $\paren {\sqrt {a x + b} }^m$
Category:Primitives‎
Category:Primitives involving Root of a x + b‎
Primitives of Functions involving Root of a x + b and Root of p x + q

== Theorem ==
This page gathers together the primitives of some functions involving $\sqrt {a x + b}$ and $\sqrt {p x + q}$.
=== Primitive of Reciprocal of $\sqrt {a x + b} \sqrt {p x + q}$ ===
=== Primitive of $x$ over $\sqrt {a x + b} \sqrt {p x + q}$ ===
=== Primitive of $\sqrt {a x + b} \sqrt {p x + q}$ ===
=== Primitive of $\sqrt {p x + q}$ over $\sqrt {a x + b}$ ===
=== Primitive of Reciprocal of $\paren {p x + q} \sqrt {\paren {a x + b} \paren {p x + q} }$ ===
Category:Primitives‎
Category:Primitives involving Root of a x + b and Root of p x + q‎
Primitives of Functions involving Root of a x + b and p x + q

== Theorem ==
This page gathers together the primitives of some functions involving $\sqrt{a x + b}$ and $p x + q$.
=== Primitive of $p x + q$ over $\sqrt{a x + b}$ ===
=== Primitive of Reciprocal of $\left({p x + q}\right) \sqrt{a x + b}$ ===
=== Primitive of $\sqrt{a x + b}$ over $p x + q$ ===
=== Primitive of $\left({p x + q}\right)^n \sqrt{a x + b}$ ===
=== Primitive of Reciprocal of $\left({p x + q}\right)^n \sqrt{a x + b}$ ===
=== Primitive of $\left({p x + q}\right)^n$ over $\sqrt{a x + b}$ ===
=== Primitive of $\sqrt{a x + b}$ over $\left({p x + q}\right)^n$ ===
Category:Primitives‎
Category:Primitives involving Root of a x + b and p x + q‎
Primitives of Functions involving a x + b and p x + q

== Theorem ==
This page gathers together the primitives of some rational functions involving $a x + b$ and $p x + q$.
=== Primitive of Reciprocal of $\left({a x + b}\right) \left({p x + q}\right)$ ===
=== Primitive of $x$ over $\left({a x + b}\right) \left({p x + q}\right)$ ===
=== Primitive of Reciprocal of $\left({a x + b}\right)^2 \left({p x + q}\right)$ ===
=== Primitive of $x$ over $\left({a x + b}\right)^2 \left({p x + q}\right)$ ===
=== Primitive of $x^2$ over $\left({a x + b}\right)^2 \left({p x + q}\right)$ ===
=== Primitive of Reciprocal of $\left({a x + b}\right)^m \left({p x + q}\right)^n$ ===
=== Primitive of $a x + b$ over $p x + q$ ===
== Primitive of $\left({a x + b}\right)^m$ over $\left({p x + q}\right)^n$ ==
Category:Primitives
Category:Primitives of Rational Functions
Category:Primitives involving a x + b and p x + q‎
Primitives of Functions of Inverse Trigonometric Functions

== Theorem ==
=== Primitive of Function of Arcsine ===
=== Primitive of Function of Arccosine ===
=== Primitive of Function of Arctangent ===
=== Primitive of Function of Arccotangent ===
=== Primitive of Function of Arcsecant ===
=== Primitive of Function of Arccosecant ===
Category:Primitives of Inverse Trigonometric Functions
Primitives of Hyperbolic Functions

== Theorem ==
This page gathers together primitives of hyperbolic functions.
In the below, $C$ is an arbitrary constant throughout.
=== Primitive of Hyperbolic Sine Function ===
=== Primitive of Hyperbolic Cosine Function ===
=== Primitive of Hyperbolic Tangent Function ===
=== Primitive of Hyperbolic Cotangent Function ===
=== Hyperbolic Secant Function: Arcsine Form ===
=== Hyperbolic Secant Function: Arctangent of Exponential Form ===
=== Primitive of Hyperbolic Cosecant Function: Logarithm Form ===
=== Primitive of Hyperbolic Cosecant Function: Hyperbolic Tangent Form ===
=== Primitive of Hyperbolic Cosecant Function: Inverse Hyperbolic Cotangent Form ===
=== Primitive of Hyperbolic Cosecant Function: Inverse Hyperbolic Cotangent of Hyperbolic Cosine Form ===
Category:Primitives
Category:Primitives of Hyperbolic Functions
Primitives of Rational Functions involving Power of a x + b

== Theorem ==
This page gathers together the primitives of some rational functions involving a general power of $a x + b$.
=== Primitive of $\paren {a x + b}^n$ ===
=== Primitive of $x \paren {a x + b}^n$ ===
=== Primitive of $x^2 \paren {a x + b}^n$ ===
== Primitive of $x^m \paren {a x + b}^n$ ==
== Also see ==
* Primitives of Rational Functions involving $a x + b$
* Primitives of Rational Functions involving $\paren {a x + b}^2$
* Primitives of Rational Functions involving $\paren {a x + b}^3$
Category:Primitives
Category:Primitives of Rational Functions
Category:Primitives involving a x + b‎
Primitives of Rational Functions involving a x + b

== Theorem ==
This page gathers together the primitives of some rational functions involving $a x + b$.
=== Primitive of Reciprocal of $a x + b$ ===
=== Primitive of $x$ over $a x + b$ ===
=== Primitive of $x^2$ over $a x + b$ ===
=== Primitive of $x^3$ over $a x + b$ ===
=== Primitive of Reciprocal of $x$ by $a x + b$ ===
=== Primitive of Reciprocal of $x^2$ by $a x + b$ ===
=== Primitive of Reciprocal of $x^3$ by$a x + b$ ===
== Also see ==
* Primitives of Rational Functions involving $\left({a x + b}\right)^2$
* Primitives of Rational Functions involving $\left({a x + b}\right)^3$
* Primitives of Rational Functions involving $\left({a x + b}\right)^n$
Category:Primitives
Category:Primitives of Rational Functions
Category:Primitives involving a x + b‎
Primitives of Rational Functions involving a x + b cubed

== Theorem ==
This page gathers together the primitives of some rational functions involving $\paren {a x + b}^3$.
=== Primitive of Reciprocal of $\paren {a x + b}^3$ ===
=== Primitive of $x$ over $\paren {a x + b}^3$ ===
=== Primitive of $x^2$ over $\paren {a x + b}^3$ ===
=== Primitive of $x^3$ over $\paren {a x + b}^3$ ===
=== Primitive of Reciprocal of $x$ by $\paren {a x + b}^3$ ===
=== Primitive of Reciprocal of $x^2$ by $\paren {a x + b}^3$ ===
=== Primitive of Reciprocal of $x^3$ by $\paren {a x + b}^3$ ===
== Also see ==
* Primitives of Rational Functions involving $a x + b$
* Primitives of Rational Functions involving $\paren {a x + b}^2$
* Primitives of Rational Functions involving $\paren {a x + b}^n$
Category:Primitives
Category:Primitives of Rational Functions
Category:Primitives involving a x + b‎
Primitives of Rational Functions involving a x + b squared

== Theorem ==
This page gathers together the primitives of some rational functions involving $\left({a x + b}\right)^2$.
=== Primitive of Reciprocal of $\left({a x + b}\right)^2$ ===
=== Primitive of $x$ over $\left({a x + b}\right)^2$ ===
=== Primitive of $x^2$ over $\left({a x + b}\right)^2$ ===
=== Primitive of $x^3$ over $\left({a x + b}\right)^2$ ===
=== Primitive of Reciprocal of $x$ by $\left({a x + b}\right)^2$ ===
=== Primitive of Reciprocal of $x^2$ by $\left({a x + b}\right)^2$ ===
=== Primitive of Reciprocal of $x^3$ by$\left({a x + b}\right)^2$ ===
== Also see ==
* Primitives of Rational Functions involving $a x + b$
* Primitives of Rational Functions involving $\left({a x + b}\right)^3$
* Primitives of Rational Functions involving $\left({a x + b}\right)^n$
Category:Primitives
Category:Primitives of Rational Functions
Category:Primitives involving a x + b‎
Primitives of Trigonometric Functions

== Theorem ==
This page gathers together primitives of trigonometric functions.
In the below, $C$ is an arbitrary constant throughout.
=== Primitive of Sine Function ===
=== Primitive of Cosine Function ===
=== Primitive of Tangent Function: Cosine Form ===
=== Primitive of Tangent Function: Secant Form ===
=== Primitive of Cotangent Function ===
=== Primitive of Secant Function: Secant plus Tangent Form ===
=== Primitive of Secant Function: Tangent plus Angle Form ===
=== Primitive of Cosecant Function: Cosecant plus Cotangent Form ===
=== Primitive of Cosecant Function: Cosecant minus Cotangent Form ===
=== Primitive of Cosecant Function: Tangent Form ===
Category:Primitives
Category:Primitives of Trigonometric Functions
Primitives which Differ by Constant

== Theorem ==
Let $F$ be a primitive for a real function $f$ on the closed interval $\closedint a b$.
Let $G$ be a real function defined on $\closedint a b$.
Then $G$ is a primitive for $f$ on $\closedint a b$ :
:$\exists c \in \R: \forall x \in \closedint a b: \map G x = \map F x + c$
That is, $F$ and $G$ differ by a constant on the whole interval.
=== Corollary ===
== Proof ==
=== Necessary Condition ===
Suppose $G$ is a primitive for $f$.
Then $F - G$ is continuous on $\closedint a b$, differentiable on $\openint a b$, and for any $x \in \openint a b$, we have:
From Zero Derivative implies Constant Function it follows that $F - G$ is constant on $\closedint a b$.
Hence the result.
=== Sufficient Condition ===
Now suppose $\map G x = \map F x + c$.
We compute:
Hence $G$ is also a primitive for $f$.
== Sources ==
* : $\S 14$: Indefinite Integrals: Definition of an Indefinite Integral
* : $\S 13.11$
* 
* 
* : $\S 16$: Indefinite Integrals: Definition of an Indefinite Integral
Category:Primitives which Differ by Constant
Category:Primitives
Primitives which Differ by Constant/Corollary

== Theorem ==
Let $f$ be an integrable function on the closed interval $\closedint a b$.
Then there exist an uncountable number of primitives for $f$ on $\closedint a b$.
== Proof ==
By definition of integrable function, $f$ has a primitive $F$ (at least one).
By Primitives which Differ by Constant, for every real number $C$, if $\map F x$ is a primitive of $f$, then so is $\map G x$ where:
:$\forall x \in \closedint a b: \map G x = \map F x + c$
The Real Numbers are Uncountable.
Hence the result.
Category:Primitives which Differ by Constant
Prince Rupert's Cube

== Prince Rupert's Cube ==
Let $C$ be a unit cube.
The largest square tunnel that can be made in $C$ through which a larger cube may be passed has sides of length:
:$\dfrac {3 \sqrt 2} 4 = 1 \cdotp 06066 \, 0$
== Proof ==
== Historical Note ==
== Sources ==
* 
* 
* 
* : $1,06066 0172 \ldots$
* : $1 \cdotp 060 \, 660 \ldots$
* : Prince Rupert's Cube: $127$
* : $1 \cdotp 06066 \, 0 \ldots$
Category:Classic Problems
Category:Solid Geometry
Prince Rupert's Cube/Historical Note

== Historical Note on Prince Rupert's Cube ==
According to , the puzzle now known as Prince Rupert's Cube was first posed by in $1693$.
The correct answer was determined by .
An incorrect solution to this puzzle, often quoted in the literature, was provided by himself, who assumed that the tunnel in question would be parallel to the space diagonal of the cube.
This provides a solution of $\sqrt 6 - \sqrt 2 \approx 1 \cdotp 03527$.
== Sources ==
* : $1 \cdotp 060 \, 660 \ldots$
* : Prince Rupert's Cube
* : $1 \cdotp 06066 \, 0 \ldots$
Category:Historical Notes
Principal Ideal Domain cannot have Infinite Strictly Increasing Sequence of Ideals

== Theorem ==
Let $\struct {D, +, \circ}$ be a principal ideal domain.
Then $D$ cannot have an infinite sequence of ideals $\sequence {j_n}_{n \mathop \in \N}$ such that:
:$\forall n \in \N: J_n \subsetneq j_{n + 1}$
== Proof ==
Let $K = \ds \bigcup_{n \mathop \in \N} J_n$.
Then from Increasing Union of Sequence of Ideals is Ideal, $K$ is an ideal of $D$.
We have that $D$ is a principal ideal domain.
Hence there exists $a \in D$ such that:
:$K = \ideal a$
where $\ideal a$ is the principal ideal of $D$ generated by $a$.
But $a \in J_m$ for some $m \in \N$.
Thus $K \subseteq J_m$
Thus it follows that $J_{m + 1} \subseteq J_m$ which contradicts our initial assertion that:
:$\forall n \in \N: J_n \subsetneq j_{n + 1}$
Hence the result.
== Sources ==
* : Chapter $9$: Rings: Exercise $17$
Category:Principal Ideal Domains
Principal Ideal Domain fulfills Ascending Chain Condition

== Theorem ==
Let $R$ be a principal ideal domain.
Then $R$ fulfils the ascending chain condition.
== Proof ==
Let $I_1 \subseteq I_2 \subseteq I_3 \subseteq \dotsb$ be an ascending chain of ideals.
Build $\ds I = \bigcup_{i \mathop = 1}^\infty I_i$. 
$I$ is an ideal.
Since $R$ is a principal ideal domain, $I = \ideal a$ for some $a \in R$.
Now, since $a \in I$, there is some $n$ such that $a \in I_n$.
Thus:
:$\ideal a \subseteq I_n$
By definition $I_n \subset I = \ideal a$, and so $I_n = I$.
Thus:
:$\forall m \ge n: I_m = I$
Category:Ideal Theory
Category:Principal Ideal Domains
Principal Ideal Domain is Bézout Domain

== Theorem ==
Let $D$ be a principal ideal domain.
Then $D$ is a Bézout domain.
== Proof ==
Let $J_1, J_2$ be ideals of $D$.
From Sum of Ideals is Ideal, $J_1 + J_2$ is an ideal of $D$.
By definition of principal ideal domain, $J_1 + J_2$ is principal.
Hence $D$ is a Bézout domain.
Category:Bézout Domains
Category:Principal Ideal Domains
Principal Ideal Domain is Dedekind Domain

== Theorem ==
Let $D$ be a principal ideal domain which is specifically not a field.
Then $D$ is a Dedekind domain.
== Proof ==
By definition of principal ideal domain $D$ is an integral domain.
By Principal Ideal Domain is Noetherian $D$ is noetherian.
By Principal Ideal Domain is Integrally Closed $D$ is integrally closed.
By Prime Ideal of Principal Ideal Domain is Maximal $D$ has Krull dimension $\le 1$.
By Integral Domain has Dimension Zero iff Field and since $D$ is not a field, the Krull dimension of $D$ is $1$.
Hence $D$ is a Dedekind domain by definition.
Category:Principal Ideal Domains
Category:Dedekind Domains
Principal Ideal Domain is Integrally Closed

== Theorem ==
Let $A$ be a principal ideal domain.
Then $A$ is integrally closed.
== Proof ==
By Principal Ideal Domain is Unique Factorization Domain $A$ is a unique factorization domain.
By Unique Factorization Domain is Integrally Closed $A$ is integrally closed.
Category:Principal Ideal Domains
Principal Ideal Domain is Unique Factorization Domain

== Theorem ==
Every principal ideal domain is a unique factorization domain.
== Proof ==
From Element of Principal Ideal Domain is Finite Product of Irreducible Elements, each element which is neither $0$ nor a unit of a principal ideal domain has a factorization of irreducible elements.
== Sources ==
* : $\S 62$. Factorization in an integral domain
Category:Integral Domains
Category:Principal Ideal Domains
Category:Unique Factorization Domains
Category:Ideal Theory
Category:Factorization
Principal Ideal from Element in Center of Ring

== Theorem ==
Let $\struct {R, +, \circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.
Let $b \in R$ be in the center of $R$.
Then:
:$\ideal b = R \circ b = \set {x \circ b: x \in R}$
where $\ideal b$ is the principal ideal generated by $b$.
== Proof ==
Let $J = R \circ b$
The center of $R$ is defined as:
:$\map Z R = \set {x \in R: \forall s \in R: s \circ x = x \circ s}$
Therefore:
:$R \circ b = b \circ R = \set {x \circ b: x \in R} = \set {b \circ x: x \in R}$
and so:
:$x \in J \implies x \circ b \in J \land b \circ x \in J$
Thus $J$ is an ideal of $R$ and so $J = \ideal b$.
== Sources ==
* : Chapter $\text {IV}$: Rings and Fields: $22$. New Rings from Old: Theorem $22.5$
Category:Principal Ideals of Rings
Principal Ideal in Integral Domain generated by Power Plus One is Subset of Principal Ideal generated by Power

== Theorem ==
Let $\struct {D, +, \circ}$ be an integral domain whose unity is $1_D$.
Let $a \in D$ be a proper element of $D$.
Then:
:$\forall n \in \Z_{\ge 0}: \ideal {a^{n + 1} } \subsetneq \ideal {a_n}$
where $\ideal x$ denotes the principal ideal of $D$ generated by $x$.
== Proof ==
We have:
It remains to be shown that $\ideal {a^{n + 1} } \ne \ideal {a^n}$.
 $\ideal {a^{n + 1} } = \ideal {a^n}$.
Then:
That is, $a$ is a unit of $D$.
This contradicts the assertion that $a$ is a proper element of $D$.
The result follows by Proof by Contradiction.
== Sources ==
* : Chapter $9$: Rings: Exercise $7$
Category:Integral Domains
Category:Principal Ideals of Rings
Principal Ideal is Ideal

== Theorem ==
Let $\struct {R, +, \circ}$ be a ring with unity.
Let $a \in R$.
Let $\ideal a$ be the principal ideal of $R$ generated by $a$.
Then $\ideal a$ is an ideal of $R$.
== Proof ==
First we establish that $\ideal a$ is an ideal of $R$, by verifying the conditions of Test for Ideal.
$\ideal a \ne \O$, as $1_R \circ a = a \in \ideal a$.
	
Let $x, y \in \ideal a$.
Then:
Let $s \in \ideal a, x \in R$.
and similarly $s \circ x \in \ideal a$.
Thus by Test for Ideal, $\ideal a$ is an ideal of $R$.
== Sources ==
* : Chapter $\text {IV}$: Rings and Fields: $22$. New Rings from Old
* : $\S 59.$ Principal ideals in a commutative ring with a one: $\S 59.1$ 
Category:Ideal Theory
Principal Ideal is Smallest Ideal

== Theorem ==
Let $\struct {R, +, \circ}$ be a ring with unity.
Let $a \in R$.
Let $\ideal a$ be the principal ideal of $R$ generated by $a$.
Let $J$ be an ideal of $R$ such that $a \in J$.
Then $\ideal a \subseteq J$.
That is, $\ideal a$ is the smallest ideal of $R$ to which $a$ belongs.
== Proof ==
Let $J$ be an ideal of $R$ such that $a \in J$.
By the definition of an ideal:
:$\forall r, s \in R: r \circ a \circ s \in J$
Also, $J$ is a group under $+$.
So every element of $\ideal a$ is in $J$.
Thus $\ideal a \subseteq J$.
== Sources ==
* : $\S 59.$ Principal ideals in a commutative ring with a one: $\S 59.1$
Category:Principal Ideals of Rings
Principal Ideal of Characteristic of Ring is Subset of Kernel of Multiple Function

== Theorem ==
Let $\struct {R, +, \circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.
Let the characteristic of $R$ be $p$.
For any $a \in R$, we define the mapping $g_a: \Z \to R$ from the integers into $R$ as:
:$\forall n \in \Z: \map {g_a} n = n \cdot a$
Then:
:$\ideal p \subseteq \map \ker {g_a}$
where:
:$\map \ker {g_a}$ is the kernel of $g_a$
:$\ideal p$ is the principal ideal of $\Z$ generated by $p$.
== Proof ==
We have from Multiple Function on Ring is Homomorphism that $g_a$ is a group homomorphism.
By definition of kernel:
:$x \in \map \ker {g_a} \iff \map {g_a} x = 0_R$
Hence to show that $\ideal p \subseteq \map \ker {g_a}$, we need to show that:
:$\forall x \in \ideal p: \map {g_a} x = 0_R$
By definition of characteristic, $p \in \Z_{\ge 0}$ is such that $\ideal p$ is the kernel of $g_1$:
:$\map {g_1} n = n \cdot 1_R$
So:
Then:
So:
:$x \in \map \ker {g_a}$
and so: 
:$\ideal p \subseteq \map \ker {g_a}$
== Sources ==
* : Chapter $\text {IV}$: Rings and Fields: $24$. The Division Algorithm: Theorem $24.8 \ 1^\circ$
Category:Homomorphism from Integers into Ring with Unity