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Definition:Euclid's Definitions - Book XI/18 - Cone
When, one side of those about the right angle in a right-angled triangle remaining fixed, the triangle is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a '''cone'''.<br>And, if the straight line which remains fixed be equal to the remaining side about the right angle which is carried round, the cone will be '''right-angled'''; if less, '''obtuse-angled'''; and if greater, '''acute-angled'''.
Definition:Euclid's Definitions - Book XI/19 - Axis of Cone
The '''axis of the cone''' is the straight line which remains fixed and about which the triangle is turned.
Definition:Euclid's Definitions - Book XI/20 - Base of Cone
And the '''base''' is the circle described by the straight line which is carried round.
Definition:Euclid's Definitions - Book XI/21 - Cylinder
When, one side of those about the right angle in a rectangular parallelogram remaining fixed, the parallelogram is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a '''cylinder'''.
Definition:Euclid's Definitions - Book XI/22 - Axis of Cylinder
The '''axis of the cylinder''' is the straight line which remains fixed and about which the parallelogram is turned.
Definition:Euclid's Definitions - Book XI/23 - Base of Cylinder
And the '''bases''' are the circles described by the two sides opposite to one another which are carried round.
Definition:Euclid's Definitions - Book XI/24 - Similar Cones and Cylinders
'''Similar cones and cylinders''' are those in which the axes and the diameters of the bases are proportional.
Definition:Euclid's Definitions - Book XI/25 - Cube
A '''cube''' is a solid figure contained by six equal squares.
Definition:Euclid's Definitions - Book XI/26 - Octahedron
An '''octahedron''' is a solid figure contained by eight equal and equilateral triangles.
Definition:Euclid's Definitions - Book XI/27 - Icosahedron
An '''icosahedron''' is a solid figure contained by twenty equal and equilateral triangles.
Definition:Euclid's Definitions - Book XI/28 - Dodecahedron
A '''dodecahedron''' is a solid figure contained by twelve equal, equilateral, and equiangular pentagons.
Definition:Conjugate of Quadratic Irrational
Let $\alpha = r + s \sqrt n$ be a quadratic irrational. Then its '''conjugate''' is defined as: :$\tilde \alpha = r - s \sqrt n$ Thus $\alpha$ and $\tilde \alpha$ are known as '''conjugate quadratic irrationals'''. Notation may vary.
Definition:Pentagon
A '''pentagon''' is a polygon with exactly $5$ sides: :300px
Definition:Hexagon
A '''hexagon''' is a polygon with exactly $6$ sides. :300px
Definition:Equimultiples
Let $a$ and $b$ be numbers. Then '''equimultiples''' of $a$ and $b$ are numbers of the form $c a$ and $c b$ where $c$ is also a number. That is, they are products arising from the multiplications of two numbers by the same common number.
Definition:Ratio
Let $x$ and $y$ be quantities which have the same dimensions. Let $\dfrac x y = \dfrac a b$ for two numbers $a$ and $b$. Then the '''ratio of $x$ to $y$''' is defined as: :$x : y = a : b$ It explicitly specifies how many times the first number contains the second.
Definition:Archimedean Property
Let $\struct  {S, \circ}$ be a closed algebraic structure on which there exists either an ordering or a norm. Let $\cdot: \Z_{>0} \times S \to S$ be the operation defined as: :$m \cdot a = \begin{cases} a & : m = 1 \\ a \circ \paren {\paren {m - 1} \cdot a} & : m > 1 \end {cases}$ === Archimedean Property on Norm === {{:Definition:Archimedean Property/Norm}} === Archimedean Property on Ordering === {{:Definition:Archimedean Property/Ordering}}
Definition:Alternate Ratio
Let $a : b$ and $c : d$ where $a : b$ denotes the ratio of $a$ to $b$. Then the '''alternate ratio''' is defined as $a : c$ and $b : d$.  :''{{:Definition:Euclid's Definitions - Book V/12 - Alternate Ratio}}'' {{EuclidDefRefNocat|V|12|Alternate Ratio}}
Definition:Ratio Ex Aequali
 :''{{:Definition:Euclid's Definitions - Book V/17 - Ratio Ex Aequali}}'' {{EuclidDefRefNocat|V|17|Ratio Ex Aequali}}
Definition:Proportion/Perturbed
Let $a, b, c$ and $A, B, C$ be magnitudes. $a, b, c$ are '''in perturbed proportion''' to $A, B, C$ : :$a : b = B : C$ :$b : c = A : B$  :''{{:Definition:Euclid's Definitions - Book V/18 - Perturbed Proportion}}'' {{EuclidDefRefNocat|V|18|Perturbed Proportion}}
Definition:Duplicate Ratio
Let $a, b, c$ be magnitudes such that: :$a : b = b : c$ Then $a$ has the '''duplicate ratio''' to $c$ of the ratio it has to $b$. That is: :$a : c$ is the '''duplicate ratio''' of $a : b$. From the definition of ratio: {{begin-eqn}} {{eqn | l=a : b       | r=b : c       | c= }} {{eqn | ll=\implies       | l=\dfrac a b       | r=\dfrac b c       | c= }} {{eqn | ll=\implies       | l=\dfrac a c       | r=\dfrac a b \dfrac b c       | c= }} {{eqn | ll=\implies       | l=\dfrac a c       | r=\left({\dfrac a b}\right)^2       | c= }} {{end-eqn}} That is: : $a : c = \left({a : b}\right)^2$  :''{{Definition:Euclid's Definitions - Book V/9 - Duplicate Ratio}} {{EuclidDefRefNocat|V|9|Duplicate Ratio}}
Definition:Triplicate Ratio
Let $a, b, c, d$ be magnitudes such that: :$a : b = b : c = c : d$ Then $a$ has the '''triplicate ratio''' to $d$ of the ratio it has to $b$. That is: :$a : d$ is the '''triplicate ratio''' of $a : b$.  :''{{Definition:Euclid's Definitions - Book V/10 - Triplicate Ratio}} {{EuclidDefRefNocat|V|10|Triplicate Ratio}}
Definition:Corresponding Magnitudes
Let $R_1 = a : b$ and $R_2 = c : d$ be ratios. Then the '''corresponding magnitudes''' of $a$ and $b$ in $R_2$ are $c$ and $d$ respectively.  :''{{Definition:Euclid's Definitions - Book V/11 - Corresponding Magnitudes}} {{EuclidDefRefNocat|V|11|Corresponding Magnitudes}}
Definition:Inverse Ratio
Let $R = a : b$ be a ratio. Then the '''inverse ratio''' to $R$ is $b : a$.  :''{{Definition:Euclid's Definitions - Book V/13 - Inverse Ratio}} {{EuclidDefRefNocat|V|13|Inverse Ratio}}
Definition:Composition of Ratio
Let $R = a : b$ be a ratio. Then the '''composition of $R$''' is the ratio $a + b : b$.  :''{{Definition:Euclid's Definitions - Book V/14 - Composition of Ratio}} {{EuclidDefRefNocat|V|14|Composition of Ratio}}
Definition:Separation of Ratio
Let $R = a : b$ be a ratio. Then the '''separation of $R$''' is the ratio $a - b : b$.  :''{{Definition:Euclid's Definitions - Book V/15 - Separation of Ratio}} {{EuclidDefRefNocat|V|15|Separation of Ratio}}
Definition:Conversion of Ratio
Let $R = a : b$ be a ratio. Then the '''conversion of $R$''' is the ratio $a : a - b$.  :''{{Definition:Euclid's Definitions - Book V/16 - Conversion of Ratio}} {{EuclidDefRefNocat|V|16|Conversion of Ratio}}
Definition:Section of Line by Line
Let a geometrical line $A$ cross over (or intersect) another line $B$. The point where they cross is called the '''section''' of the $B$ by $A$ (or equivalently, of $A$ by $B$).
Definition:Similar Figures
Two rectilineal figures are '''similar''' : :They have corresponding angles, all of which are equal :They have corresponding sides, all of which are proportional.
Definition:Reciprocal Relation
 :''{{Definition:Euclid's Definitions - Book VI/2 - Reciprocally Related Figures}} {{EuclidDefRefNocat|VI|2|Reciprocally Related Figures}}
Definition:Similar Triangles
Similar triangles are triangles whose corresponding angles are the same, but whose corresponding sides may be of different lengths. :300px Thus $\triangle ABC$ is similar to $\triangle DEF$: :$\angle ABC = \angle EFD$ :$\angle BCA = \angle EDF$ :$\angle CAB = \angle DEF$
Definition:Rectification
'''Rectification''' is an archaic term meaning '''the process of finding length''', specifically in the context of arcs.
Definition:Reciprocal Proportion
Let $P$ and $Q$ be geometric figures of the same type (that is, having the same number and configuration of sides). Let $A$ and $B$ be sides of $P$, and let $C$ and $D$ be sides of $Q$, such that $A$ and $C$ are corresponding sides, and $B$ and $D$ also be corresponding sides. Then $P$ and $Q$ have sides which are '''in reciprocal proportion''', or are '''reciprocally proportional''', if: :$A : D = B : C$ where $A : D$ is the ratio of the lengths of $A$ and $D$.
Definition:Compound Ratio
Let $R_1 = a : b$ and $R_2 = c : d$ be ratios. The '''compound ratio''' of $R_1$ and $R_2$ is defined as: :$R_1 R_2 = a c : b d$ That is, it is the product of the two ratios.
Definition:Minimal Polynomial
Let $L / K$ be a field extension. Let $\alpha \in L$ be algebraic over $K$. === Definition 1 === {{Definition:Minimal Polynomial/Definition 1}} === Definition 2 === {{Definition:Minimal Polynomial/Definition 2}} === Definition 3 === {{Definition:Minimal Polynomial/Definition 3}}
Definition:Algebraic Closure
Let $K$ be a field. An '''algebraic closure''' of $K$ is an algebraically closed algebraic field extension of $K$. An '''algebraic closure''' of $K$ can be denoted $\overline K$.
Definition:Normal Extension
=== Definition 1 === {{:Definition:Normal Extension/Definition 1}} === Definition 2 === {{:Definition:Normal Extension/Definition 2}}
Definition:Separable Extension
Let $K$ be a field. Let $L/K$ be an algebraic field extension. Then $L/K$ is a '''separable extension'''  every $\alpha\in L$ is separable over $K$.
Definition:Galois Extension/Finite
Let $L/K$ be a finite field extension. === Definition 1 === {{:Definition:Galois Extension/Finite/Definition 1}} === Definition 2 === {{:Definition:Galois Extension/Finite/Definition 2}} === Definition 3 === {{:Definition:Galois Extension/Finite/Definition 3}}
Definition:Intermediate Field
Let $L / K$ be a field extension. Let $F$ be a field such that: : $K \subset F \subset L$ Then $F$ is an '''intermediate field'''.
Definition:Embedding (Galois Theory)
Let $K$ and $L$ be fields. A (field) monomorphism $\phi: K \to L$ is called an '''embedding''' of $K$ in $L$.
Definition:Inclusion-Reversing Mapping
Let $A, B$ be sets of sets and $\phi: A \to B$ be a mapping. Then $\phi$ is '''inclusion-reversing''' : : for every pair of sets $a_1, a_2 \in A$ such that $a_1 \subseteq a_2$: :: $\phi \left({a_2}\right) \subseteq \phi \left({a_1}\right)$
Definition:Galois Group of Field Extension
Let $L / K$ be a field extension. The '''Galois group''' of $L / K$ is the subgroup of the automorphism group of $L$ consisting of field automorphisms that fix $K$ point-wise: :$\Gal {L / K} = \set{\sigma \in \Aut L: \forall k \in K: \map \sigma k = k}$ === As a topological group === The notation $\Gal {L / K}$ is also a shorthand for the topological group: :$\struct {\Gal {L / K}, \tau}$ where $\tau$ is the Krull topology. === Alternative Definition === More generally, we can abandon the condition that $L / K$ be Galois if we choose an algebraic closure $\overline K$ such that $L \subseteq \overline K$ and define: :$\Gal {L / K} = \left\{ {\sigma: L \to \overline K: \sigma}\right.$ is an embedding of $L$ such that $\sigma$ fixes $K$ point-wise$\left.\right\}$ This set will form a group  $L / K$ is normal.
Definition:Gradation Compatible with Ring Structure
Let $\left({M, \cdot}\right)$ be a semigroup. Let $\left({R, +, \circ}\right)$ be a ring. Let $(R_n)_{n\in M}$ be an gradation of type $M$ on the additive group of $R$. The gradation is '''compatible with the ring structure'''  :$\forall m, n \in M : \forall x \in S_m, y \in S_n: x \circ y \in S_{m \cdot n}$ and so: :$S_m S_n \subseteq S_{m\cdot n}$
Definition:Filtered Algebra
A '''filtered algebra''' is a generalization of the notion of a graded algebra. A '''filtered algebra''' over the field $k$ is an algebra $\left({A_k, \oplus}\right)$ over $k$ which has an increasing sequence $\left\{{0}\right\} \subset F_0 \subset F_1 \subset \cdots \subset F_i \subset \cdots \subset A$ of substructures of $A$ such that: :$\displaystyle A = \bigcup_{i \mathop \in \N} F_i$ and that is compatible with the multiplication in the following sense:   :$\forall m, n \in \N: F_m \cdot F_n \subset F_{n+m}$
Definition:Graded Algebra
Let $R$ be a graded commutative ring with unity. An algebra $A$ over $R$ is a '''graded algebra'''. That is, a '''graded algebra''' is an algebra over a ring where the ring is a graded ring.
Definition:Quadratic Algebra
A '''quadratic algebra''' $A$ is a filtered algebra whose generator consists of degree one elements, with defining relations of degree 2. A '''quadratic algebra''' $A$ is determined by a vector space of generators $V = A_1$ and a subspace of homogeneous quadratic relations $S \subseteq V \times V$. Thus : :$A = T \left({V}\right) / \left \langle {S}\right \rangle$ and inherits its grading from the tensor algebra $T \left({V}\right)$. If the subspace of relations may also contain inhomogeneous degree 2 elements, $S \subseteq k \times V \times \left({V \times V}\right)$, this construction results in a '''filtered quadratic algebra'''. A graded quadratic algebra $A$ as above admits a '''quadratic dual''': the quadratic algebra generated by $V^*$ and with quadratic relations forming the orthogonal complement of $S$ in $V^* \times V^*$.
Definition:Division Algebra
Let $\left({A_F, \oplus}\right)$ be an algebra over field $F$ such that $A_F$ does not consist solely of the zero vector $\mathbf 0_A$ of $A_F$. === Definition 1 === {{:Definition:Division Algebra/Definition 1}} === Definition 2 === {{:Definition:Division Algebra/Definition 2}}
Definition:Real Algebra
A '''real algebra''' is an algebra over a field where the field in question is the field of real numbers $\R$.
Definition:Associative Algebra
Let $R$ be a commutative ring. Let $\left({A_R, *}\right)$ be an algebra over $R$. Then $\left({A_R, *}\right)$ is an '''associative algebra'''  $*$ is an associative operation. That is: :$\forall a, b, c \in A_R: \left({a * b}\right) * c = a * \left({b * c}\right)$
Definition:Unital Algebra
Let $R$ be a commutative ring. Let $\left({A, *}\right)$ be an algebra over $R$.  Then $\left({A, *}\right)$ is a '''unital algebra'''  the algebraic structure $\left({A, \oplus}\right)$ has an identity element. That is: :$\exists 1_A \in A: \forall a \in A: a * 1_A = 1_A * a = a$
Definition:Quadratic Real Algebra
A '''quadratic real algebra''' is a quadratic algebra over the field of real numbers.
Definition:Normed Division Algebra
Let $\left({A_F, \oplus}\right)$ be a unitary algebra where $A_F$ is a vector space over a field $F$. Then $\left({A_F, \oplus}\right)$ is a normed divison algebra iff $A_F$ is a normed vector space such that: :$\forall a, b \in A_F: \left \Vert{a \oplus b}\right \Vert = \left \Vert{a}\right \Vert \left \Vert{b}\right \Vert$ where $\left \Vert{a}\right \Vert$ denotes the norm of $a$.
Definition:Unitary Division Algebra
Let $\left({A_F, \oplus}\right)$ be a division algebra. Then $\left({A_F, \oplus}\right)$ is a '''unitary division algebra'''  it has an identity element $1_A$ called a '''unit''' for $\oplus$: :$\exists 1_A \in A_R: \forall a \in A_R: a \oplus 1_A = 1_A \oplus a = a$ The unit is usually denoted $1$ when there is no source of confusion with the identity elements of the underlying structures of the algebra.
Definition:Unit of Algebra
Let $R$ be a commutative ring. Let $\left({A_R, \oplus}\right)$ be a unitary algebra over $R$. The '''unit''' of $\left({A_R, \oplus}\right)$, denoted $1_A$, is the identity element of the operation $\oplus$: :$\forall a \in A_R: a \oplus 1_A = 1_A \oplus a = a$ It is sometimes referred to as the '''multiplicative identity''' of $\left({A_R, \oplus}\right)$. It is usually denoted $1$ when there is no source of confusion with the identity elements of the underlying structures of the algebra.
Definition:Multiplicative Inverse
Let $\strut {A_F, \oplus}$ be a unitary algebra whose unit is $1$ and whose zero is $0$. Let $a \in A_F$ such that $a \ne 0$. A '''multiplicative inverse''' of $a$ is an element $b \in A_F$ such that: :$a \oplus b = 1 = b \oplus a$ === Field === {{:Definition:Multiplicative Inverse/Field}} === Multiplicative Inverse of Number === {{:Definition:Multiplicative Inverse/Number}}
Definition:Power-Associative Operation
Let $\circ$ be a binary operation. Then $\circ$ is defined as being '''power-associative on $S$''' : :$\forall x \in S: \paren {x \circ x} \circ x = x \circ \paren {x \circ x}$
Definition:Alternative Operation
Let $\circ$ be a binary operation. Then $\circ$ is defined as being '''alternative on $S$''' : :$\forall T := \set {x, y} \subseteq S: \forall x, y, z \in T: \paren {x \circ y} \circ z = x \circ \paren {y \circ z}$ That is, $\circ$ is associative over any two elements of $S$. For example, for any $x, y \in S$: :$\paren {x \circ y} \circ x = x \circ \paren {y \circ x}$ :$\paren {x \circ x} \circ y = x \circ \paren {x \circ y}$ and so on.
Definition:Multilinear Mapping
Let $\left({R, +_R, \times_R}\right)$ be a commutative ring. Let $\left({A_1, +_1, \circ_1}\right)_R, \left({A_2, +_2, \circ_2}\right)_R, \ldots, \left({A_n, +_n, \circ_n}\right)_R, \left({A_{n+1}, +_{n+1}, \circ_{n+1}}\right)_R$ be $R$-modules. Let $\oplus: A_1 \times A_2 \times \cdots \times A_n \to A_{n+1}$ be a multiary operator with the property that: $\forall \left({a_1, a_2, \ldots, a_n}\right) \in A_1 \times A_2 \times \cdots \times A_n$: * $a_1 \mapsto a_1 \oplus a_2 \oplus \cdots \oplus a_n$ is a linear transformation from $A_1$ to $A_{n+1}$ * $a_2 \mapsto a_1 \oplus a_2 \oplus \cdots \oplus a_n$ is a linear transformation from $A_2$ to $A_{n+1}$ * $\vdots$ * $a_n \mapsto a_1 \oplus a_2 \oplus \cdots \oplus a_n$ is a linear transformation from $A_n$ to $A_{n+1}$ Then $\oplus$ is a '''multilinear mapping'''.
Definition:Associator
Let $\left({A_R, \oplus}\right)$ be an algebra over a ring. Consider the trilinear mapping $\left[{\cdot, \cdot, \cdot}\right]: A_R^3 \to A_R$ defined as: :$\forall a, b, c \in A_R: \left[{a, b, c}\right] := \left({a \oplus b}\right) \oplus c - a \oplus \left({b \oplus c}\right)$ Then $\left[{\cdot, \cdot, \cdot}\right]$ is known as the '''associator''' of $\left({A_R, \oplus}\right)$. It can be considered a measure of how much associativity of $\oplus$ fails in $\left({A_R, \oplus}\right)$. Note that trivially if $\left({A_R, \oplus}\right)$ is an associative algebra, then: :$\forall a, b, c \in A_R: \left[{a, b, c}\right] = \mathbf 0_R$
Definition:Commutator
The '''commutator''' of an algebraic structure can be considered a measure of how commutative the structure is. === Groups === {{:Definition:Commutator/Group}} === Rings === {{:Definition:Commutator/Ring}} === Algebras === {{:Definition:Commutator/Algebra}}
Definition:Commutative Algebra
Let $R$ be a commutative ring. Let $\struct {A_R, \oplus}$ be an algebra over $R$. Then $\struct {A_R, \oplus}$ is a commutative algebra  $\oplus$ is a commutative operation. That is: :$\forall a, b \in A_R: a \oplus b = b \oplus a$
Definition:Alternating Bilinear Mapping
Let $\left({A_R, \oplus}\right)$ be an algebra over a ring. By definition, $\oplus$ is a bilinear mapping. Then $\oplus$ is an '''alternating bilinear mapping''' : :$\forall a \in A_R: a \oplus a = 0$ For rings with characteristic other than two, the following condition is equivalent: :$\forall a, b \in A_R: a \oplus b = - b \oplus a$
Definition:Commutator/Group
Let $\struct {G, \circ}$ be a group. Let $g, h \in G$. The '''commutator''' of $g$ and $h$ is the operation: :$\sqbrk {g, h} := g^{-1} \circ h^{-1} \circ g \circ h$
Definition:Commutator/Ring
Let $\struct {R, +, \circ}$ be a ring. Let $a, b \in R$. The '''commutator''' of $a$ and $b$ is the operation: :$\sqbrk {a, b} := a \circ b + \paren {-b \circ a}$ or more compactly: :$\sqbrk {a, b} := a \circ b - b \circ a$
Definition:Commutator/Algebra
Let $\struct {A_R, \oplus}$ be an algebra over a ring. Consider the bilinear mapping $\sqbrk {\, \cdot, \cdot \,}: A_R^2 \to A_R$ defined as: :$\forall a, b \in A_R: \sqbrk {a, b} := a \oplus b - b \oplus a$ Then $\sqbrk {\, \cdot, \cdot \,}$ is known as the '''commutator''' of $\struct {A_R, \oplus}$. Note that trivially if $\struct {A_R, \oplus}$ is a commutative algebra, then: :$\forall a, b \in A_R: \sqbrk {a, b} = \mathbf 0_R$
Definition:Star-Algebra
Let $A = \left({A_F, \oplus}\right)$ be an unitary division algebra. Let $A$ have a mapping $*: A \to A$ such that: :$\forall a \in A: \left({a^*}\right)^* = a$ :$\forall a, b \in A: \left({a \oplus b}\right)^* = b^* \oplus a^*$ Then $A$ is a '''$*$-algebra''' (usually voiced '''star-algebra'''). The mapping $*: A \to A$ is a conjugation on $A$.
Definition:Antihomomorphism
Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be a mapping from one algebraic structure $\struct {S, \circ}$ to another $\struct {T, *}$. Then $\phi$ is an '''antihomomorphism''' : :$\forall x, y \in S: \map \phi {x \circ y} = \map \phi y * \map \phi x$ For structures with more than one operation, $\phi$ may be '''antihomomorphic''' for a subset of those operations.
Definition:Antihomomorphism/Ring Antihomomorphism
Let $\struct {R, +, \circ}$ and $\struct {S, \oplus, *}$ be rings. Then $\phi: R \to S$ is a '''ring antihomomorphism''' : {{begin-eqn}} {{eqn | lo= \forall a, b \in R:       | l = \map \phi {a + b}       | r = \map \phi a \oplus \map \phi b }} {{eqn | lo= \forall a, b \in R:       | l = \map \phi {a \circ b}       | r = \map \phi b * \map \phi a }} {{end-eqn}}
Definition:Conjugation (Abstract Algebra)
Let $A = \left({A_F, \oplus}\right)$ be an algebra over a field $F$. Let $C: A_F \to A_F$ be a mapping such that: :$\forall a \in A: C \left({C \left({a}\right)}\right) = a$ :$\forall a, b \in A: C \left({a \oplus b}\right) = C \left({b}\right) \oplus C \left({a}\right)$ Then $C$ is called a '''conjugation''' on $A$. === Conjugate === {{:Definition:Conjugation (Abstract Algebra)/Conjugate}}
Definition:Nicely Normed Star-Algebra
Let $A = \left({A_F, \oplus}\right)$ be a star-algebra whose conjugation is denoted $*$. Then $A$ is a '''nicely normed $*$-algebra''' : :$\forall a \in A: a + a^* \in \R$ :$\forall a \in A, a \ne 0: 0 < a \oplus a^* = a^* \oplus a \in \R$ === Real Part === Let $a \in A$ be an element of a '''nicely normed $*$-algebra'''. Then the '''real part''' of $a$ is given by: :$\Re \left({a}\right) = \dfrac {a + a^*} 2$   === Imaginary Part === Let $a \in A$ be an element of a '''nicely normed $*$-algebra'''. Then the '''imaginary part''' of $a$ is given by: :$\Im \left({a}\right) = \dfrac {a - a^*} 2$ === Norm === Let $a \in A$ be an element of a '''nicely normed $*$-algebra'''. Then we can define a norm on $a$ by: :$\left\Vert{a}\right\Vert^2 = a \oplus a^*$
Definition:Real Star-Algebra
Let $A = \left({A_F, \oplus}\right)$ be an star-algebra whose conjugation is denoted $*$. Then $A$ is a '''real $*$-algebra''' : :$\forall a \in A: a^*= a$
Definition:Subalgebra
Let $R$ be a commutative ring. Let $\left({A, *}\right)$ be an algebra over $R$. A '''subalgebra''' of $A$ is a submodule $B \subseteq A$ such that: :$\forall x, y \in B: x * y \in B$ That is, such that $B$ is closed under $*$.
Definition:Unital Subalgebra
Let $R$ be a commutative ring. Let $\left({A_R, *}\right)$ be an unital algebra over $R$ whose unit is $1_A$. Let $\left({B_R, *}\right)$ be a subalgebra of $A_R$. === Definition 1 === {{:Definition:Unital Subalgebra/Definition 1}} === Definition 2 === {{:Definition:Unital Subalgebra/Definition 2}}
Definition:Generator of Field
Let $F$ be a field. Let $S \subseteq F$ be a subset and $K \le F$ a subfield. The '''field generated by $S$''' is the smallest subfield of $F$ containing $S$. The '''subring of $F$ generated by $K \cup S$''', written $K \left[{S}\right]$, is the smallest subring of $F$ containing $K \cup S$. The '''subfield of $F$ generated by $K \cup S$''', written $K \left({S}\right)$, is the smallest subfield of $F$ containing $K \cup S$.
Definition:Generator of Semigroup
Let $\left({S, \circ}\right)$ be a semigroup. Let $\varnothing \subset X \subseteq S$. Let $\left({T, \circ}\right)$ be the smallest subsemigroup of $\left({S, \circ}\right)$ such that $X \subseteq T$. Then: : $X$ is a '''generator'''  of $\left({T, \circ}\right)$ : $X$ '''generates''' $\left({T, \circ}\right)$ : $\left({T, \circ}\right)$ is the '''subsemigroup of $\left({S, \circ}\right)$ generated by $X$'''. This is written $T = \left \langle {X} \right \rangle$.