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7. Notation

General:=Means a definition of something. For a example,X:={(x,y):x2+y2=1}Logics∀xFor all x∃xExists x∃!xExists unique xa∧ba and ba∨bEither a or b or both¬aNot aSetsX∈SX is a seta∈Ba is in a set BA⊆BA is a subset of BA⊂BA is a proper subset of BA∩BIntersection of A and BA∪BUnion of A and BA∖BDifference of sets A and B∣A∣Number of elements in a set AX×YCartesian product of sets X and YXYInner product of sets X and YxY{x}YX=∑iXi{Xi} is a partition of Xmin⁡x∈Xf(x)Minimum of f(x) over the set Xarg min⁡x∈Xf(x)m∈X such that m=min⁡x∈Xf(x)Binary relations∼Equivalence relationX∈S∼X is a set with equivalence relation[x]∼Equivalence class containing x≤PPartial order≤TTotal orderX∈S≤PX is a set with partial orderX∈S≤TX is a set with total orderY∈C(X)Y is a chain in XMappingsf:A→Bf is a map from set A to set Bf:x↦yf maps element x to element yf:A→1−1Bf−injectivef:A→B,f(A)=Bf−surjectivef:A↔Bf−bijectiveGroups(G,+)Additive group(G,⋅)Multiplicative groupG∈G(+)G is an additive group G∈G(⋅)G is a multiplicative group G∈GAG is an abelian group G∈G↺G is a cyclic group H⊆GMH is a subgroup of group MH⊂GMH is a proper subgroup of group MH⊲GMH is a normal subgroup of group M⟨g1,…,gn⟩GA group generated by elements g1,…,gn⟨S⟩GA group generated by elements of a set Sord aOrder of element aϕ(n)Euler functionSnGroup of permutations of a set {1,…,n}Sn↺A subset of cycles in a permutations group Snσ,τ∈Sn↺,σ∩τ=∅Disjoint cycles[G:H]Subgroup index of H in GM/HFactor group of M relative to Hϕ:A⇝GBf is a group homomorphismA⇝ϕGBf is a group homomorphismϕ:A≅GBf is a group isoomorphismA≅ϕGBf is a group isomorphismker⁡ϕHomomorphism kernelZnAdditive group of integers modulo nRingsA⊆RSA is a subring of SI⊲RNI is ideal in Nchar ACharacteristic of a ring of Ap∈P(A)p is a prime in Au∈A∗u is a unit in Af∈A−f is irreducible element in Af∈A+f is reducible element in AI∈PI(A)I is a prime ideal in AI∈PI(A)I is a primary ideal in AI∈MI(A)I is a maximal ideal in AIRadial of ideal IM/HFactor ring of M over ideal H(I:J)Ideal quotient⟨S⟩RA ring generated by elements of a set S⟨a1,…,an⟩IAn ideal generated by elements a1,…,an⟨S⟩IAn ideal generated by elements of a set Sϕ:A⇝RBϕ is a ring homomorphismA⇝ϕRBf is a ring homomorphismϕ:A≅RBϕ is a ring isomorphismA≅ϕRBf is a ring isomorphismdim⁡ADimension of a ring Aht PHeight of a prime ideal Pker⁡ϕHomomorphism kernelA[x1,…,xn]A ring of polynomials over x1,…,xn variables with coefficients in AA[α1,…,αn]A minimal ring containing A,α1,…,αna∣ba dibives ba∈P(A)a is a prime in Agcd⁡(a,b)Greatest common divisor of a and bdeg⁡pPolynomial degreeLT(p)Leading term of polynomial pLC(p)Leading coefficent of polynomial ppϕAction of ϕ on coefficents of polynomial pϕxHomomorphism extended from R1⇝ϕRR2 to R1[x]⇝ϕxRR2[x]IB(A)Integral closure of A in Bv∈V(F)v− discrete valuation over field FS−1RRing of fractions BPRing localization (at ideal)BaRing localization (at point)⨁Bidirect sum of BiRings taxonomyR∈RR is a ringR∈R1R is a ring with identityA∈RCA is a commutative ringB∈RIDB is an integral domainD∈RGCDD is a GCD domainU∈RUFDP is a unique factorization domainP∈RPIDP is a principal ideal domainE∈REP is a euclidean domainL∈RLL is a local ringN∈RNN is a noetherean ringA∈RAA is an Artin ringV∈RVV is a valuation ringD∈RDVRD is a discrete valuation ringB∈RGRB is a graded ringModulesM∈MRM is an R-moduleN⊆MKN is a submodule of KX∈⊥L(M)X is linear independent in MX∈BM(N)X is a basis for module N⟨a1,…,an⟩MRR-module generated by elements a1,…,an⟨S⟩MRR-module generated by elements of a set Sϕ:A⇝MBϕ is a module homomorphismA⇝ϕMBf is a module homomorphismϕ:A≅MBϕ is a module isomorphismA≅ϕMBf is a module isomorphismdim⁡RM≡rankRMModule dimension/rank⨁Midirect sum of MiAlgebrasL∈Aϕ,AL is A-algebra with homomorphism ϕϕ:A⇝ABϕ is a algebra homomorphismA⇝ϕABf is a algebra homomorphismϕ:A≅ABϕ is a algebra isomorphismA≅ϕABf is a algebra isomorphismFieldsF∈FF is a fieldF⊆FEF is a subfield of EE/FE is a field extension of FF(B)Field of fractions over integral domain B∼FEquivalence relation in a field of fractionsabEquivalence class in a fiel d of fractionsE/FE is a field extension of F[E:F]Extension degree of E over FF(A)Minimal field containing F and AF(α)Simple extensionF(x1,…,xn)Function fieldF‾EAlgebraic closure of F in EF‾Algebraic closure of FpolF(α)Minimal polynomial of α over the field Fdeg⁡αDegree of element αordα(p)Multiplicity of root α for pP∥FA set of polynomial splits in FF(∥P)Splitting field for polynomials P over FT∈⊥A(F)T is algebraically independent over FB∈BFtr(E)B is a transcendence basis in E over FtrdegF(E)Transcendence degree of E over FEndF(E)Field endomorphisms in EAutF(E)Field automorphisms in Eϕ:E1⇝∣FE2F-homomorphismϕ:E1≅∣FE2F-isomorphismϕ:E≅∣FEF-automorphismGal(E/F)Galois group of E over FFixS(E)Fixed subfield by SρFrobenius endomorphismF∈FPF is a perfect fieldField extensionsE/AFAlgebraic extensionE/TFTranscendental extensionE/⊲FNormal extensionE/⊟FSeparable extensionE/□FPurely inseparable extensionE/GalFGalois extension\def\arraystretch{1.5} \begin{array}{ll} \textbf{General} \\ := & \text{Means a definition of something. } \text{For a example}, X:= \{(x, y): x^2+y^2=1\} \\ \\ \textbf{Logics} \\ \forall x & \text{For all } x \\ \exists x & \text{Exists } x \\ \exists! x & \text{Exists unique } x \\ a \wedge b & a \text{ and } b \\ a \vee b & \text{Either } a \text{ or } b \text{ or both} \\ \neg a & \text{Not } a \\ \\ \textbf{Sets} \\ X \in \mathcal S & X \text{ is a set} \\ a \in B & a \text{ is in a set } B\\ A \subseteq B & A \text{ is a subset of } B\\ A \subset B & A \text{ is a proper subset of } B\\ A \cap B & \text{Intersection of } A \text{ and }B\\ A \cup B & \text{Union of } A \text{ and }B\\ A \setminus B &\text{Difference of sets } A \text{ and } B\\ |A| & \text{Number of elements in a set } A \\ X \times Y &\text{Cartesian product of sets } X \text{ and } Y \\ XY &\text{Inner product of sets } X \text{ and } Y \\ xY &\{x\}Y\\ X=\sum_iX_i &\{X_i\} \text{ is a partition of }X\\ \min_{x \in X} f(x) & \text{Minimum of } f(x) \text{ over the set }X \\ \argmin_{x \in X} f(x) & m \in X \text{ such that } m = \min_{x \in X} f(x) \\ \\ \textbf{Binary relations} \\ \sim & \text{Equivalence relation} \\ X \in \mathcal S_{\sim} & X \text{ is a set with equivalence relation} \\ [x]_{\sim} & \text{Equivalence class containing } x \\ \le_P & \text{Partial order}\\ \le_T & \text{Total order}\\ X \in \mathcal S_{\le_P} & X \text{ is a set with partial order} \\ X \in \mathcal S_{\le_T} & X \text{ is a set with total order} \\ Y \in \mathfrak C(X) & Y \text{ is a chain in } X \\ \\ \textbf{Mappings} \\ f:A \to B & f \text{ is a map from set } A \text{ to set } B\\ f: x \mapsto y & f \text{ maps element } x \text{ to element } y\\ f: A \overset{1-1}{\to} B & f - \text{injective} \\ f:A \to B, f(A) = B & f - \text{surjective} \\ f: A \leftrightarrow B & f - \text{bijective} \\ \\ \textbf{Groups} \\ (G, +) & \text{Additive group} \\ (G, \cdot) & \text{Multiplicative group} \\ G \in \mathcal{G}_{(+)} & G\text{ is an additive group } \\ G \in \mathcal{G}_{(\cdot)} & G\text{ is a multiplicative group } \\ G \in \mathcal{G}^{\mathcal A} & G\text{ is an abelian group } \\ G \in \mathcal{G}^{\circlearrowleft} & G\text{ is a cyclic group } \\ H \subseteq_G M & H\text{ is a subgroup of group } M \\ H \subset_G M & H\text{ is a proper subgroup of group } M \\ H \lhd_G M & H\text{ is a normal subgroup of group } M \\ \lang g_1, \ldots , g_n \rang_G & \text{A group generated by elements } g_1, \ldots , g_n\\ \lang S \rang_G & \text{A group generated by elements of a set } S\\ \text{ord }a & \text{Order of element } a \\ \phi(n) & \text{Euler function} \\ S_n & \text{Group of permutations of a set } \{1, \ldots, n\} \\ S_n^\circlearrowleft & \text{A subset of cycles in a permutations group } S_n\\ \sigma, \tau \in S_n^\circlearrowleft, \sigma \cap \tau = \empty & \text{Disjoint cycles} \\ [G:H] & \text{Subgroup index of } H \text{ in } G \\ M/H & \text{Factor group of } M \text{ relative to } H \\ \phi: A \rightsquigarrow_G B & f \text{ is a group homomorphism}\\ A \overset{\phi}{\rightsquigarrow}_G B & f \text{ is a group homomorphism}\\ \phi: A \cong_G B & f \text{ is a group isoomorphism}\\ A \overset{\phi}{\cong}_G B & f \text{ is a group isomorphism}\\ \ker \phi & \text{Homomorphism kernel} \\ \Z_n &\text{Additive group of integers modulo } n \\ \\ \textbf{Rings} \\ A \subseteq_R S & A \text{ is a subring of } S\\ I \lhd_R N& I \text{ is ideal in } N \\ \text{char }A & \text{Characteristic of a ring of } A\\ p \in \mathfrak P(A)& p \text{ is a prime in } A \\ u \in A^* & u \text{ is a unit in }A \\ f \in A^- & f \text{ is irreducible element in } A\\ f \in A^+ & f \text{ is reducible element in } A\\ I \in \mathfrak P_I(A)& I \text{ is a prime ideal in } A \\ I \in \sqrt \mathfrak P_I(A)& I \text{ is a primary ideal in } A \\ I \in \mathfrak M_I(A) & I \text{ is a maximal ideal in } A \\ \sqrt I & \text{Radial of ideal }I\\ M/H & \text{Factor ring of } M \text{ over ideal } H \\ (I:J) & \text{Ideal quotient} \\ \lang S \rang_R & \text{A ring generated by elements of a set } S\\ \lang a_1, \ldots , a_n \rang_I & \text{An ideal generated by elements } a_1, \ldots , a_n\\ \lang S \rang_I & \text{An ideal generated by elements of a set } S\\ \phi: A \rightsquigarrow_R B & \phi \text{ is a ring homomorphism}\\ A \overset{\phi}{\rightsquigarrow}_R B & f \text{ is a ring homomorphism}\\ \phi: A \cong_R B & \phi \text{ is a ring isomorphism}\\ A \overset{\phi}{\cong}_R B & f \text{ is a ring isomorphism}\\ \dim A &\text{Dimension of a ring }A\\ \text{ht } P & \text{Height of a prime ideal } P\\ \ker \phi & \text{Homomorphism kernel} \\ A[x_1, \ldots , x_n] &\text{A ring of polynomials over }x_1, \ldots , x_n \text{ variables with coefficients in } A \\ A[\alpha_1, \ldots , \alpha_n] &\text{A minimal ring containing } A, \alpha_1, \ldots , \alpha_n \\ a \mid b & a \text{ dibives } b\\ a \in \mathfrak P(A) & a \text{ is a prime in } A\\ \gcd(a,b) & \text{Greatest common divisor of } a \text{ and } b \\ \deg p & \text{Polynomial degree} \\ LT(p) & \text{Leading term of polynomial }p \\ LC(p) & \text{Leading coefficent of polynomial }p \\ p^{\phi} & \text{Action of } \phi \text{ on coefficents of polynomial }p \\ \phi_x & \text{Homomorphism extended from } R_1 \overset{\phi}\rightsquigarrow_R R_2 \text{ to }R_1[x] \overset{\phi_x}\rightsquigarrow_R R_2[x] \\ \mathfrak I_B(A) & \text{Integral closure of }A \text{ in } B\\ v \in \mathcal V(F) & v - \text{ discrete valuation over field } F \\ S^{-1}R & \text{Ring of fractions }\\ B_P & \text{Ring localization (at ideal)}\\ B_a & \text{Ring localization (at point)}\\ \bigoplus B_i &\text{direct sum of }B_i \\ \\ \textbf{Rings taxonomy} \\ R \in \mathcal{R} & R \text{ is a ring} \\ R \in \mathcal{R^1} & R \text{ is a ring with identity} \\ A \in \mathcal R^{\mathcal {C}} & A \text{ is a commutative ring} \\ B \in \mathcal R^{\mathcal {ID}} & B \text{ is an integral domain} \\ D \in \mathcal{R}^\mathcal{GCD} & D \text{ is a GCD domain} \\ U \in \mathcal{R}^\mathcal{UFD} & P \text{ is a unique factorization domain} \\ P \in \mathcal{R}^\mathcal{PID} & P \text{ is a principal ideal domain} \\ E \in \mathcal{R}^\mathcal{E} & P \text{ is a euclidean domain} \\ L \in \mathcal{R}^{\mathcal L} & L \text{ is a local ring} \\ N \in \mathcal{R}^\mathcal{N} & N \text{ is a noetherean ring} \\ A \in \mathcal{R}^{\mathcal A} & A \text{ is an Artin ring} \\ V \in \mathcal{R}^{\mathcal V} & V \text{ is a valuation ring} \\ D \in \mathcal{R}^{\mathcal {DVR}} & D \text{ is a discrete valuation ring} \\ B \in \mathcal{R}^{\mathcal {GR}} & B \text{ is a graded ring} \\ \\ \textbf{Modules} \\ M \in \mathcal{M}_R & M \text{ is an }R\text{-module} \\ N \subseteq_M K & N \text{ is a submodule of } K\\ X \in \bot_L(M) & X \text{ is linear independent in } M \\ X \in \mathfrak B_M(N) & X \text{ is a basis for module }N \\ \lang a_1, \ldots , a_n \rang_{M_R} & R\text{-module generated by elements } a_1, \ldots , a_n\\ \lang S \rang_{M_R} & R\text{-module generated by elements of a set } S\\ \phi: A \rightsquigarrow_M B & \phi \text{ is a module homomorphism}\\ A \overset{\phi}{\rightsquigarrow}_M B & f \text{ is a module homomorphism}\\ \phi: A \cong_M B & \phi \text{ is a module isomorphism}\\ A \overset{\phi}{\cong}_M B & f \text{ is a module isomorphism}\\ \dim_RM \equiv \text{rank}_R M & \text{Module dimension/rank} \\ \bigoplus M_i &\text{direct sum of }M_i \\ \\ \textbf{Algebras} \\ L \in \mathcal A_{\phi, A} & L \text{ is }A\text{-algebra with homomorphism } \phi \\ \phi: A \rightsquigarrow_A B & \phi \text{ is a algebra homomorphism}\\ A \overset{\phi}{\rightsquigarrow}_A B & f \text{ is a algebra homomorphism}\\ \phi: A \cong_A B & \phi \text{ is a algebra isomorphism}\\ A \overset{\phi}{\cong}_A B & f \text{ is a algebra isomorphism}\\ \\ \textbf{Fields} \\ F \in \mathcal{F} & F \text{ is a field} \\ F \subseteq_F E & F \text{ is a subfield of } E \\ E / F & E \text{ is a field extension of } F \\ \mathfrak F(B) & \text{Field of fractions over integral domain } B \\ \sim_F & \text{Equivalence relation in a field of fractions} \\ \frac{a}{b} & \text{Equivalence class in a fiel d of fractions} \\ E/F & E\text{ is a field extension of }F \\ [E:F] & \text{Extension degree of } E \text{ over } F \\ F(A) & \text{Minimal field containing }F \text{ and } A \\ F(\alpha) & \text{Simple extension} \\ F(x_1, \ldots, x_n) & \text{Function field} \\ \overline F_E & \text{Algebraic closure of }F \text{ in } E \\ \overline F & \text{Algebraic closure of }F \\ \mathfrak{pol}_F(\alpha) & \text{Minimal polynomial of }\alpha \text{ over the field }F \\ \deg \alpha & \text{Degree of element } \alpha \\ \text{ord}_{\alpha}(p) &\text{Multiplicity of root }\alpha \text{ for }p\\ P \parallel F &\text{A set of polynomial splits in }F\\ F(\parallel P) &\text{Splitting field for polynomials } P \text{ over }F\\ T \in \bot_A(F) & T \text{ is algebraically independent over } F \\ B \in \mathfrak B^{tr}_F(E) & B\text{ is a transcendence basis in }E \text{ over } F\\ \text{trdeg}_F(E) & \text{Transcendence degree of }E \text{ over }F\\ \text{End}_F(E) & \text{Field endomorphisms in } E \\ \text{Aut}_F(E) & \text{Field automorphisms in } E \\ \phi: E_1 \rightsquigarrow_{|F}E_2 &F\text{-homomorphism} \\ \phi: E_1 \cong_{|F}E_2 &F\text{-isomorphism} \\ \phi: E \cong_{|F}E &F\text{-automorphism} \\ \text{Gal}(E/F) & \text{Galois group of }E \text{ over }F \\ \text{Fix}_S(E) & \text{Fixed subfield by }S \\ \rho & \text{Frobenius endomorphism} \\ F \in \mathcal F^{\mathcal P} & F \text{ is a perfect field} \\ \\ \textbf{Field extensions} \\ E/_A F &\text{Algebraic extension} \\ E/_T F &\text{Transcendental extension} \\ E/_\lhd F &\text{Normal extension} \\ E/_\boxminus F &\text{Separable extension} \\ E/_\Box F &\text{Purely inseparable extension} \\ E/_\text{Gal} F &\text{Galois extension} \\ \end{array}