hcju/proofwikips huggingface dataset:
Definition:Functor Category
Let $C$ and $D$ be categories. The '''functor category''' $\operatorname{Funct}(C, D)$ is the category with: {{DefineCategory |ob = covariant functors $C \to D$ |mor = natural transformations |comp = vertical composition of natural transformations |id = identity natural transformations }} {{definition wanted|the contravariant case}}
Definition:Generated Subgroup
Let $G$ be a group. Let $S \subset G$ be a subset. === Definition 1 === {{Definition:Generated Subgroup/Definition 1}} === Definition 2 === {{Definition:Generated Subgroup/Definition 2}} === Definition 3 === {{Definition:Generated Subgroup/Definition 3}}
Definition:Generated Subgroup/Definition 1
The '''subgroup generated by $S$''' is the smallest subgroup containing $S$.
Definition:Generated Subgroup/Definition 3
Let $S^{-1}$ be the set of inverses of $S$. The '''subgroup generated by $S$''' is the set of words on the union $S \cup S^{-1}$.
Definition:Generator of Ring
Let $R$ be a ring. Let $S \subset R$ be a subset. Then $S$ is a '''generator''' of $R$  $R$ is the subring generated by $S$.
Definition:Generator of Ring Extension
Let $f : A \to B$ be a ring extension of commutative rings with unity. Let $S \subset B$ be a subset. Then $S$ is a '''generator of $B$ over $A$'''  $B$ is the ring extension generated by $S$.
Definition:Generated Normal Subgroup
Let $G$ be a group. Let $S \subset G$ be a subset. === Definition 1 === {{Definition:Generated Normal Subgroup/Definition 1}} === Definition 2 === {{Definition:Generated Normal Subgroup/Definition 2}} === Definition 3 === {{Definition:Generated Normal Subgroup/Definition 3}}
Definition:Generated Normal Subgroup/Definition 1
The '''normal subgroup generated by $S$''', denoted $\gen {S^G}$,  is the intersection of all normal subgroups of $G$ containing $S$.
Definition:Generated Normal Subgroup/Definition 2
The '''normal subgroup generated by $S$''', denoted $\gen {S^G}$, is the subgroup generated by the set of conjugates of $S$: :$S^G = \set {g^{−1}sg: g \in G, s \in S}$
Definition:Generator of Ideal of Ring
Let $R$ be a commutative ring.  Let $I \subset R$ be an ideal. Let $S \subset I$ be a subset. Then $S$ is a '''generator''' of $I$  $I$ is the ideal generated by $S$.
Definition:Riemann Zeta Function/Zero/Nontrivial
The '''nontrivial zeroes''' of the Riemann zeta function $\zeta$ are the zeroes of $\zeta$ that are not trivial.
Definition:Formal Derivative of Dirichlet Series
Let $(a_n) : \N \to \C$ be an arithmetic function. Let $F(s)=\displaystyle \sum_{n\mathop=1}^\infty\frac{a_n}{n^s}$ be the corresponding formal Dirichlet series. The '''formal derivative''' of $F(s)$ is the formal Dirichlet series: :$\displaystyle \sum_{n\mathop=1}^\infty\frac{a_n\cdot \log n}{n^s}$
Definition:Abscissa of Convergence
Let $f \left({s}\right)$ be a Dirichlet series The '''abscissa of convergence''' of $f$ is the extended real number $\sigma_0 \in \overline \R$ defined by: :$\sigma_0 = \displaystyle \inf \left\{ {\operatorname{Re} \left({s}\right) : s \in \C, f \left({s}\right) \text{converges} }\right\}$ where $\inf \varnothing = +\infty$.
Definition:Dirichlet Inverse of Arithmetic Function
Let $f : \N \to \C$ be an arithmetic function. Let $\varepsilon$ be the identity arithmetic function. A '''Dirichlet inverse''' of $f$ is an arithmetic function $g$ such that $g*f = \varepsilon$, where $*$ denotes Dirichlet convolution. That is, an inverse of $f$ in the ring of arithmetic functions.
Definition:Formal Product of Dirichlet Series
Let $f,g : \N \to \C$ be arithmetic functions. Let $F,G$ be their formal Dirichlet series. The '''formal product''' of $F$ and $G$ is the formal Dirichlet series: :$\displaystyle \sum_{n \mathop = 1}^\infty \frac{(f*g)(n)}{n^s}$ where $*$ denotes Dirichlet convolution.
Definition:Ring of Arithmetic Functions
Let $\AA$ be the set of all arithmetic functions. Let $*$ denote Dirichlet convolution, and $+$ the pointwise sum of functions. The '''ring of arithmetic functions''' is the ring $\struct {\AA, +, *}$.
Definition:Consistent (Logic)/Set of Formulas/Propositional Logic/Definition 1
Then $\FF$ is '''consistent for $\mathscr P$''' : :There exists a logical formula $\phi$ such that $\FF \not \vdash_{\mathscr P} \phi$ That is, some logical formula $\phi$ is '''not''' a $\mathscr P$-provable consequence of $\FF$.
Definition:Consistent (Logic)/Set of Formulas/Propositional Logic/Definition 2
Suppose that in $\mathscr P$, the Rule of Explosion (Variant 3) holds. Then $\FF$ is '''consistent for $\mathscr P$''' : :For every logical formula $\phi$, not ''both'' of $\phi$ and $\neg \phi$ are $\mathscr P$-provable consequences of $\FF$
Definition:Pole of Complex Function
Let $U \subset \C$ be an open subset. Let $f : U \to \C$ be a holomorphic function. Let $p\in \C$ be an isolated singularity of $f$. === Definition 1 === The point $p$ is a '''pole''' of $f$  the Laurent expansion of $f$ around $p$ has the form: :$f(z) = \displaystyle \sum_{k = -n}^\infty a_k(z-p)^k$ === Definition 2 === The point $p$ is a '''pole''' of $f$  there exists a natural number $m>0$ such that: :$\displaystyle\lim_{z \to p}(z-p)^m f(z) \in \C\setminus\{0\}$ === Definition 3 === The point $p$ is a '''pole''' of $f$  the improper limit: :$\displaystyle\lim_{z\to p}|f(z)| = \infty$.
Definition:Complex Disk
Let $a \in \C$ be a complex number. Let $R>0$ be a real number. === Open disk === {{Definition:Complex Disk/Open}} === Closed disk === {{Definition:Complex Disk/Closed}}
Definition:Complex Disk/Open
The '''open (complex) disk of center $a$ and radius $R$''' is the set: :$\map B {a, R} = \set {z \in \C: \cmod {z - a} < R}$ where $\cmod {\, \cdot \,}$ denotes complex modulus.
Definition:Complex Disk/Closed
The '''closed (complex) disk of center $a$ and radius $R$''' is the set: :$\map B {a, R} = \set {z \in \C: \cmod {z - a} \le R}$ where $\cmod {\, \cdot \,}$ denotes complex modulus.
Definition:Unit Ball
Let $V$ be a normed vector space with norm $\norm {\, \cdot \,}$. The '''closed unit ball''' of $V$, denoted $\operatorname {ball} V$, is the set: :$\set {v \in V: \norm v_V \mathop \le 1}$
Definition:Design
A '''design''' is an ordered pair $\tuple {X, \mathcal B}$ where: :$X$ is a non-empty finite set of points :$\mathcal B$ is a multiset of blocks.
Definition:Point (Design Theory)
Let $X$ be non-empty finite set. If $X$ is to be analyzed in the context of Design Theory, then its elements are referred to as '''points'''. In application, the '''points''' in $X$ are often variables.
Definition:Block
Let $X$ be a set of points. Let $\mathcal B$ be a multiset of subsets of $X$. Then the elements of $\mathcal B$ are called '''blocks'''.
Definition:Real Interval/Definition 2
A '''real interval''' is a subset of $\R$ that is one of the following real interval types: * closed bounded interval * open bounded interval * left half-open bounded interval * right half-open bounded interval * closed and bounded on the right, also known as a closed unbounded below real interval * open and bounded on the right, also known as an open unbounded below real interval * closed and bounded on the left, also known as a closed unbounded above real interval * open and bounded on the left, also known as an open unbounded above real interval * unbounded interval without endpoints
Definition:Real Interval Types
It is usual to define intervals in terms of inequalities. These are in the form of a pair of brackets, either round or square, enclosing the two endpoints of the interval separated by two dots. Whether the bracket at either end is round or square depends on whether the end point is '''inside''' or '''outside''' the interval, as specified in the following. Let $a, b \in \R$ be real numbers. === Bounded Intervals === ==== Open Interval ==== {{:Definition:Real Interval/Open}} ==== Half-Open Interval ==== {{:Definition:Real Interval/Half-Open}} ==== Closed Interval ==== {{:Definition:Real Interval/Closed}} ==== Bounded Interval ==== {{:Definition:Real Interval/Bounded}} === Unbounded Intervals === {{:Definition:Real Interval/Unbounded}} === Other Intervals === ==== Empty Interval ==== {{:Definition:Real Interval/Empty}} ==== Singleton Interval ==== {{:Definition:Real Interval/Singleton}} ==== Unit Interval ==== {{:Definition:Real Interval/Unit Interval}}
Definition:Real Interval/Half-Open/Right
The '''right half-open (real) interval''' from $a$ to $b$ is the subset: :$\hointr a b := \set {x \in \R: a \le x < b}$
Definition:Real Interval/Half-Open/Left
The '''left half-open (real) interval''' from $a$ to $b$ is the subset: :$\hointl a b := \set {x \in \R: a < x \le b}$
Definition:Compact Real Interval
A '''compact real interval''' is a closed interval of the form: :$[a\,.\,.\,b]$ with $a,b\in \R$.
Definition:Removable Discontinuity of Real Function
Let $A \subseteq \R$ be a subset of the real numbers. Let $f: A \to \R$ be a real function. Let $f$ be discontinuous at $a\in A$. === Definition 1  === {{Definition:Removable Discontinuity of Real Function/Definition 1}} === Definition 2  === {{Definition:Removable Discontinuity of Real Function/Definition 2}}
Definition:Removable Discontinuity of Real Function/Definition 1
The point $a$ is a '''removable discontinuity''' of $f$  the limit $\displaystyle \lim_{x \mathop \to a} \map f x$ exists.
Definition:Removable Discontinuity of Real Function/Definition 2
The point $a$ is a '''removable discontinuity''' of $f$  there exists $b \in \R$ such that the function $f_b$ defined by: :$\map {f_b} x = \begin {cases} \map f x &: x \ne a \\ b &: x = a \end {cases}$ is continuous at $a$.
Definition:Discontinuous Mapping/Real Function/Point
Let $A \subseteq \R$ be a subset of the real numbers. Let $f : A \to \R$ be a real function. Let $a\in A$. Then $f$ is '''discontinuous''' at $a$  $f$ is not continuous at $a$.
Definition:Discontinuous Mapping/Real Function
=== At a Point === {{Definition:Discontinuous Mapping/Real Function/Point}}
Definition:Discontinuous Mapping/Topological Space/Point
Let $T_1 = \left({A_1, \tau_1}\right)$ and $T_2 = \left({A_2, \tau_2}\right)$ be topological spaces. Let $f: A_1 \to A_2$ $x \in T_1$ be a mapping from $A_1$ to $A_2$. Then by definition $f$ is continuous at $x$ if for every neighborhood $N$ of $f \left({x}\right)$ there exists a neighborhood $M$ of $x$ such that $f \left({M}\right) \subseteq N$. Therefore, $f$ is discontinuous at $x$ if for some neighbourhood $N$ of $f \left({x}\right)$ and every neighbourhood $M$ of $x$, $f \left({M}\right) \nsubseteq N$. The point $x$ is called a '''discontinuity of $f$'''.
Definition:Polynomial Function/Real/Definition 1
A '''real polynomial function''' on $S$ is a function $f: S \to \R$ for which there exist: :a natural number $n\in \N$ :real numbers $a_0, \ldots, a_n \in \R$ such that for all $x \in S$: :$\map f x = \displaystyle \sum_{k \mathop = 0}^n a_k x^k$ where $\sum$ denotes indexed summation.
Definition:Polynomial Function/Complex/Definition 1
A '''complex polynomial function''' on $S$ is a function $f : S \to \C$ for which there exist: * a natural number $n\in \N$ * complex numbers $a_0, \ldots, a_n \in \C$ such that for all $z \in S$: :$\map f z = \displaystyle \sum_{k \mathop = 0}^n a_k z^k$ where $\displaystyle \sum$ denotes indexed summation.
Definition:Polynomial Function/Complex/Definition 2
Let $\C \sqbrk X$ be the polynomial ring in one variable over $\C$. Let $\C^S$ be the ring of mappings from $S$ to $\C$. Let $\iota \in \C^S$ denote the inclusion $S \hookrightarrow \C$. A '''complex polynomial function''' on $S$ is a function $f: S \to \C$ which is in the image of the evaluation homomorphism $\C \sqbrk X \to \C^S$ at $\iota$.
Definition:Polynomial Function/Real/Definition 2
Let $\R \sqbrk X$ be the polynomial ring in one variable over $\R$. Let $\R^S$ be the ring of mappings from $S$ to $\R$. Let $\iota \in \R^S$ denote the inclusion $S \hookrightarrow \R$. A '''real polynomial function''' on $S$ is a function $f: S \to \R$ which is in the image of the evaluation homomorphism $\R \sqbrk X \to \R^S$ at $\iota$.
Definition:Polynomial Function/Ring
Let $R$ be a commutative ring with unity. Let $S \subset R$ be a subset. === Definition 1 === {{Definition:Polynomial Function/Ring/Definition 1}} === Definition 2 === {{Definition:Polynomial Function/Ring/Definition 2}}
Definition:Polynomial Function/Ring/Definition 1
A '''polynomial function''' on $S$ is a mapping $f : S \to R$ for which there exist: : a natural number $n \in \N$ : $a_0, \ldots, a_n \in R$ such that for all $x\in S$: :$\map f x = \displaystyle \sum_{k \mathop = 0}^n a_k x^k$ where $\sum$ denotes indexed summation.
Definition:Polynomial Function/Ring/Definition 2
Let $R \sqbrk X$ be the polynomial ring in one variable over $R$. Let $R^S$ be the ring of mappings from $S$ to $R$. Let $\iota \in R^S$ denote the inclusion $S \hookrightarrow R$. A '''polynomial function''' on $S$ is a mapping $f : S \to R$ which is in the image of the evaluation homomorphism $R \sqbrk X \to R^S$ at $\iota$.
Definition:Polynomial in Ring Element/Definition 1
A '''polynomial in $x$ over $S$''' is an element $y \in R$ for which there exist: :a natural number $n \in \N$ :$a_0, \ldots, a_n \in S$ such that: :$y = \displaystyle \sum_{k \mathop = 0}^n a_k x^k$ where: :$\displaystyle \sum$ denotes indexed summation :$x^k$ denotes the $k$th power of $x$
Definition:Polynomial in Ring Element/Definition 2
Let $S[X]$ be the polynomial ring in one variable over $S$. A '''polynomial in $x$ over $S$''' is an element that is in the image of the evaluation homomorphism $S[X]\to R$ at $x$.
Definition:Monomial of Polynomial Ring/One Variable
Let $R \sqbrk X$ be a polynomial ring over $R$ in one indeterminate $X$. A '''monomial''' of $R \sqbrk X$ is an element that is a power of $X$.
Definition:Monomial of Polynomial Ring/Multiple Variables
Let $I$ be a set. Let $R \sqbrk {\family {X_i}_{i \mathop \in I} }$ be a polynomial ring in $I$ variables $\family {X_i}_{i \mathop \in I}$. Let $y \in R \sqbrk {\family {X_i}_{i \mathop \in I} }$. A '''monomial''' of $R \sqbrk {\family {X_i}_{i \mathop \in I} }$ is an element that is a product of variables; specifically: === Definition 1 === {{Definition:Monomial of Polynomial Ring/Multiple Variables/Definition 1}} === Definition 2 === {{Definition:Monomial of Polynomial Ring/Multiple Variables/Definition 2}}
Definition:Discontinuous Mapping/Topological Space
=== At a Point === {{Definition:Discontinuous Mapping/Topological Space/Point}}
Definition:Strong Fibonacci Pseudoprime
A '''strong Fibonacci pseudoprime''' is a Carmichael number which also satisfies one of the following conditions: === Type I === {{:Definition:Strong Fibonacci Pseudoprime/Type I}} === Type II === {{:Definition:Strong Fibonacci Pseudoprime/Type II}}
Definition:Strong Fibonacci Pseudoprime/Type I
A '''strong Fibonacci pseudoprime of type I''' is a Carmichael number $N = \displaystyle \prod p_i$ such that an even number of the prime factors $p_i$ are of the form $4 m - 1$ where: {{begin-eqn}} {{eqn | n = 1       | l = 2 \paren {p_i + 1}       | o = \divides       | r = \paren {N - 1}       | c = for those $p_i$ of the form $4 m - 1$ }} {{eqn | n = 2       | l = \paren {p_i + 1}       | o = \divides       | r = \paren {N \pm 1}       | c = for those $p_i$ of the form $4 m + 1$ }} {{end-eqn}}
Definition:Strong Fibonacci Pseudoprime/Type II
A '''strong Fibonacci pseudoprime of type II''' is a Carmichael number $N = \displaystyle \prod p_i$ such that an odd number of the prime factors $p_i$ are of the form $4 m - 1$ where: :$2 \paren {p_i + 1} \divides \paren {N - p_i}$ for all $p_i$
Definition:Cunningham Chain/First Kind
A '''Cunningham chain of the first kind''' is a (finite) sequence $\left({p_1, p_2, \ldots, p_n}\right)$ such that: :$(1): \quad \forall i \in \left\{ {1, 2, \ldots, n - 1}\right\}: p_{i + 1} = 2 p_i + 1$ :$(2): \quad p_i$ is prime for all $i \in \left\{ {1, 2, \ldots, n - 1}\right\}$ :$(3): \quad n$ is not prime such that $2 n + 1 = p_1$ :$(4): \quad 2 p_n + 1$ is not prime. Thus: : each term except the last is a Sophie Germain prime : each term except the first is a safe prime.
Definition:Aurifeuillian Factorization
An '''Aurifeuillian factorization''' is an operation to find the prime factors of integers of various forms based upon the identity: :$a^2 + b^2 = \left({a + \sqrt {2a b} + b}\right) \left({a - \sqrt{2ab} + b}\right)$
Definition:Polydivisible Number
=== Definition 1 === {{:Definition:Polydivisible Number/Definition 1}} === Definition 2 === {{:Definition:Polydivisible Number/Definition 2}} === Sequence of Polydivisible Numbers === {{:Definition:Polydivisible Number/Sequence}}
Definition:Fischer-Griess Monster
The '''Fischer-Griess Monster''' is the automorphism group of the Griess algebra.
Definition:Vigintillion
=== Short Scale === {{:Definition:Vigintillion/Short Scale}} === Long Scale === {{:Definition:Vigintillion/Long Scale}}
Definition:Number-Naming System
There are various '''number-naming systems''' for naming large numbers (that is: greater than $1 \, 000 \, 000$). === Short Scale === {{:Definition:Number-Naming System/Short Scale}} === Long Scale === {{:Definition:Number-Naming System/Long Scale}}
Definition:Number-Naming System/Short Scale
The '''short scale system''' is the number-naming system which uses: :the word '''million''' for $10^6 = 1 \, 000 \, 000$ :the Latin-derived prefixes '''bi-''', '''tri-''', '''quadri-''', '''quint-''', etc. for each further multiple of $1 \, 000$, appended to the root '''-(i)llion''', corresponding to the indices $2$, $3$, $4$, $5$, $\ldots$ Thus: {{begin-axiom}} {{axiom | lc= one billion:         | m = = 1 \, 000 \, 000 \, 000         | mm= = 10^9 = 10^{2 \times 3 + 3} }} {{axiom | lc= one trillion         | m = = 1 \, 000 \, 000 \, 000 \, 000         | mm= = 10^{12} = 10^{3 \times 3 + 3} }} {{axiom | lc= one quadrillion         | m = = 1 \, 000 \, 000 \, 000 \, 000 \, 000         | mm= = 10^{15} = 10^{4 \times 3 + 3} }} {{axiom | lc= one quintillion         | m = = 1 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000         | mm= = 10^{18} = 10^{5 \times 3 + 3} }} {{end-axiom}} Thus '''one $n$-illion''' equals $1000 \times 10^{3 n}$ or $10^{3 n + 3}$
Definition:Number-Naming System/Long Scale
The '''long scale system''' is the number-naming system which uses: :the word '''million''' for $10^6 = 1 \, 000 \, 000$ :the Latin-derived prefixes '''bi-''', '''tri-''', '''quadri-''', '''quint-''', etc. for each further multiple of $1 \, 000 \, 000$, appended to the root '''-(i)llion''', corresponding to the indices $2$, $3$, $4$, $5$, $\ldots$ Thus: {{begin-axiom}} {{axiom | lc= one billion:         | m = = 1 \, 000 \, 000 \, 000 \, 000         | mm= = 10^{12} = 10^{2 \times 6} }} {{axiom | lc= one trillion         | m = = 1 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000         | mm= = 10^{18} = 10^{3 \times 6} }} {{axiom | lc= one quadrillion         | m = = 1 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000         | mm= = 10^{24} = 10^{4 \times 6} }} {{axiom | lc= one quintillion         | m = = 1 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000         | mm= = 10^{30} = 10^{5 \times 6} }} {{end-axiom}} Thus '''one $n$-illion''' equals $10^{6 n}$. Additional terms are occasionally found to fill some of the gaps, but these are rare nowadays: {{begin-axiom}} {{axiom | lc= one milliard:         | m = = 1 \, 000 \, 000 \, 000         | mm= = 10^9 }} {{axiom | lc= one billiard         | m = = 1 \, 000 \, 000 \, 000 \, 000 \, 000         | mm= = 10^{15} }} {{end-axiom}}
Definition:Vigintillion/Also known as
Some sources use the word '''vigillion''' for '''vigintillion'''.
Definition:Vigintillion/Short Scale
'''Vigintillion''' is a name for $10^{63}$ in the short scale system: :'''One vigintillion''' $= 10^{3 \times 20 + 3}$
Definition:Vigintillion/Long Scale
'''Vigintillion''' is a name for $10^{120}$ in the long scale system: :'''One vigintillion''' $= 10^{6 \times 20}$
Definition:Multiindex Power Notation
Let $M$ be a commutative monoid. Let $I$ be a set. Let $\mathbf x = (x_i)_{i\in I}$ be a family of elements of $M$. Let $\mathbf a = (a_i)_{i\in I}$ be a family of natural numbers of finite support. We denote $\mathbf x^{\mathbf a} = \displaystyle \prod_{i \in I} x_i^{a_i}$ where: * $x_i^{a_i}$ is the $a_i$th power of $x_i$ * $\prod$ denotes product with finite support
Definition:Restriction of Scalars of Module
Let $R$ and $S$ be commutative rings with unity. Let $f : R \to S$ be a ring homomorphism. Let $M$ be an $S$-module with ring representation $\rho$. The '''restriction of scalars''' of $M$ to $R$ via $f$ is the $R$-module with ring representation $\rho \circ f$.
Definition:Centillion
=== Short Scale === {{:Definition:Centillion/Short Scale}} === Long Scale === {{:Definition:Centillion/Long Scale}}
Definition:Centillion/Short Scale
'''Centillion''' is a name for $10^{303}$ in the short scale system: :'''One centillion''' $= 10^{3 \times 100 + 3}$
Definition:Milli-Millillion/Short Scale
'''Milli-millillion''' is a name for $10^{3 \, 000 \, 003}$ in the short scale system: :'''One milli-millillion''' $= 10^{3 \times 1 \, 000 \, 000 + 3}$
Definition:Module Structure of Ring
Let $R$ be a commutative ring with unity. Then '''$R$ as an $R$-module''' is the $R$-module $R$ with ring action: :$R\times R \to R$ equal to the ring product of $R$.
Definition:Koti
'''Koti''' is a Jaina word for a '''hundred-hundred-thousand''' ($10 \, 000 \, 000$).
Definition:Pakoti
'''Pakoti''' is a Jaina word for a koti multiplied by a koti: that is: :$10 \, 000 \, 000 \times 10 \, 000 \, 000$
Definition:Asankhyeya
'''Asankhyeya''' is a Jaina word for a koti to the power of $20$: that is: :$10 \, 000 \, 000^{20} = 10^{140}$
Definition:Module Structure of Polynomial Ring
Let $R$ be a commutative ring with unity. Let $R \sqbrk X$ be a polynomial ring in one variable $X$ over $R$. The '''$R$-module structure''' on $R \sqbrk X$ is the module structure as an algebra over $R$.
Definition:Ring Representation
Let $R$ be a ring. Let $M$ be an abelian group. A '''ring representation''' of $R$ on $M$ is a ring homomorphism from $R$ to the endomorphism ring $\map {\operatorname {End} } M$. === Unital Ring Representation === {{:Definition:Ring Representation/Unital}}
Definition:Constant Polynomial/Definition 1
The polynomial $P$ is a '''constant polynomial'''  its coefficients of $x^k$ are zero for $k \ge 1$.
Definition:Constant Polynomial/Definition 2
The polynomial $P$ is a '''constant polynomial'''  $P$ is either the zero polynomial or has degree $0$.
Definition:Constant Polynomial/Definition 3
The polynomial $P$ is a '''constant polynomial'''  it is in the image of the canonical embedding $R \to R \left[{x}\right]$.
Definition:Minimal Polynomial/Definition 1
The '''minimal polynomial''' of $\alpha$ over $K$ is the unique monic polynomial $f \in K \sqbrk x$ of smallest degree such that $\map f \alpha = 0$.
Definition:Minimal Polynomial/Definition 2
The '''minimal polynomial''' of $\alpha$ over $K$ is the unique irreducible, monic polynomial $f \in K \left[{x}\right]$ such that $f \left({\alpha}\right) = 0$.
Definition:Minimal Polynomial/Definition 3
The '''minimal polynomial''' of $\alpha$ over $K$ is the unique monic polynomial $f \in K \left[{x}\right]$ that generates the kernel of the evaluation homomorphism $K \left[{x}\right] \to L$ at $\alpha$. That is, such that for all $g \in K \left[{x}\right]$: :$g \left({\alpha}\right) = 0$  $f$ divides $g$.
Definition:Integral Element of Algebra
Let $A$ be a commutative ring with unity. Let $f : A \to B$ be a commutative $A$-algebra. Let $b\in B$. === Definition 1 === {{Definition:Integral Element of Algebra/Definition 1}} === Definition 2 === {{Definition:Integral Element of Algebra/Definition 2}} === Definition 3 === {{Definition:Integral Element of Algebra/Definition 3}} === Definition 4 === {{Definition:Integral Element of Algebra/Definition 4}}
Definition:Integral Element of Algebra/Definition 1
The element $b$ is '''integral''' over $A$  it is a root of a monic polynomial in $A[x]$.
Definition:Integral Element of Algebra/Definition 2
The element $b$ is '''integral''' over $A$  the generated subalgebra $A[b]$ is a finitely generated module over $A$.
Definition:Integral Element of Algebra/Definition 3
The element $b$ is '''integral''' over $A$  the generated subalgebra $A[b]$ is contained in a subalgebra $C\leq B$ which is a finitely generated module over $A$.