Abstract algebra studies algebraic structures.

Algebraic operation $\omega: X^n \to X$ is a mapping from the a Cartesian power of a set to the set itself, where $n$ is the arity (元数) of the algebraic operation: nullary operation, $n = 0$; unary operation, $n = 1$; binary operation, $n = 2$; finitary operation, $n \in \mathbb{N}$; infinitary operation, $n = \aleph_\alpha, \alpha \in \mathbb{N}$.

Algebraic structure $((\omega_i)_{i \in I}, (R_j)_{j \in J})$ on a set is a class of finitary operations and a class of finitary relations on the set. Algebraic system $(X, (\omega_i)_{i \in I}, (R_j)_{j \in J})$ is a set endowed with an algebraic structure. Basic or primitive operations and relations of an algebraic system are those in its algebraic structure. Universal algebra (泛代数) or algebra is an algebraic system with no basic relation. Relational system or model (in logic; 模型) is an algebraic system with no basic operation.

Homomorphism (同态) is a map between two algebraic systems that preserves their basic operations and relations; in other words, it is a morphism in a category of algebraic systems: $\phi \in \text{Hom}(A, A')$. Endomorphism (自同态) is a homomorphism from an algebraic system to itself. The set of all endomorphisms on an algebraic system is denoted as $\text{End}(A)$. The classification of endomorphisms on finite-dimensional vector spaces over an algebraically closed field is call Jordan cannonical form.

## Group

Magma (原群), semigroup (半群), monoid (幺半群), group (群), and abelian group (交换群) are algebras $(X, ∗)$ where $∗$ is a binary operation satisfying a cumulative list of properties, defined in the following table.

Table: Group-like Algebras $(X, ∗)$ by Cumulative Properties of the Operation

Property Property Definition Algebra
closure $\forall a,b \in X, a ∗ b \in X$ magma
associativity $\forall a,b,c \in X, (a ∗ b) ∗ c = a ∗ (b ∗ c)$ semigroup
identity $\exists e \in X, \forall a \in X, e ∗ a = a ∗ e = a$ monoid
inverse $\forall a \in X, \exists b \in X, b ∗ a = a ∗ b = e$ group
commutativity $\forall a,b \in X, a ∗ b = b ∗ a$ abelian group

Integer group $(\mathbb{Z}, +)$ is the group consisting of integers, with the usual addition. For the integer group, its identity is 0 and the inverse of an element $a$ is $-a$. Dihedral groups have underlying sets consisting of symmetries like rotations and flections, and composition as group operation.

Symmetric group $(S(X), \circ)$ on a set $X$ is the group consisting of all bijective transformations on the set, with the composition operation. Automorphism group $(\text{Aut} X, \circ)$ on a space $(X, \dots)$ is the group consisting of all automorphisms on the space, with the composition operation. For a symmetric group or an automorphism group, its identity is the identity map $\text{id}$ and the inverse of an element $f$ is $f^{-1}$.

Left action (作用) $g \cdot x$ of a group $(G, ∗)$ on a set $X$ is a map $\phi: G \times X \mapsto X$ that satisfies identity and associativity: $1 ∗ x = x$; $g_1 \cdot (g_2 \cdot x) = (g_1 ∗ g_2) \cdot x$. Right action (右作用) of a group on a set is similarly defined, only with group elements appearing on the right.

## Ring

Ring (环) $(X, (+, ×))$ is a set endowed with two binary operations called addition $+$ and multiplication $×$ such that: the set with the addition is an abelian group $(X, +)$, the set with the multiplication is a monoid $(X, ×)$, and the multiplication left and right distributes over the addition: $a × (b + c) = (a × b) + (a × c)$, $(a + b) × c = (a × c) + (b × c)$. Subtraction $-$ is the inverse operation of addition $+$. Semiring (半环) is an algebra similar to a ring, but without additive inverses. Commutative ring (交换环) is a ring whose multiplication is commutative.

Additive identity $0$ of a ring is an identity of the addition. Additive identity is unique; the additive inverse of every element is unique. Multiplication by the additive identity annihilates a ring: $0 × a = a × 0 = 0$. Multiplicative identity $1$ of a ring is an identity of the multiplication. Multiplication by the additive inverse of a multiplicative identity equals its additive inverse: $-1 × a = -a$. Zero ring or trivial ring is a ring with the same additive and multiplicative identities: $0 = 1$. Every zero ring is a singleton consisting of its additive/multiplicative identity only. Integer ring $(\mathbb{Z}, (+, ×))$ is the ring consisting of integers, with the usual addition and multiplication. For the integer ring, its additive identity is $0$, additive inverse of an element $a$ is $-a$, and the multiplicative identity is $1$. Modular arithmetic $\mathbb{Z}/n\mathbb{Z}$ is a ring. Matrix ring $M_n(R)$ or $R_n$ over a ring is the ring consisting of all n-by-n matrices over the underlying ring, endowed with matrix addition and matrix multiplication. For a matrix ring, its additive identity is $0$ and its multiplicative identity is $I$.

### Polynomial

Polynomial of one variable over a commutative ring is a function in the form of a finite sum of products of elements in the ring and power functions with non-negative integer exponents: $f: \mathcal{R} \mapsto \mathcal{R}$, $f(x) = \sum_{k=0}^n a_k x^k$ where $(a_k)_{k=0}^n \subset \mathcal{R}$. We call $(a_k)_{k=0}^n$ coefficients and $(a_k x_k)_{k=0}^n$ a terms. Root of a polynomial is an element of its zero set. Fundamental Theorem of Algebra: Every complex polynomial has a root [@Girard1629; @Descartes1637]. Every real polynomial can be decomposed into a product of linear and quadratic real polynomials [MacLaurin and Euler; Gauss]. Multiplicity of a root of a polynomial is the number of times it appears in the factorization of the polynomial. Polynomial ring $\mathcal{R}[ x ]$ over a commutative ring $\mathcal{R}$ is the ring consisting of all polynomials in variable $x$ over the underlying ring, endowed with pointwise addition and multiplication.

Polynomial of n variables over a commutative ring is a function in the form of a finite sum of products of elements in the ring and power functions with non-negative integer exponents in each variable: $f: \mathcal{R}^n \mapsto \mathcal{R}$, $f(x_i)_{i=1}^n = \sum_{k=0}^m a_k \prod_{i=1}^n x_i^{k_i}$. The n-ary polynomial ring over a commutative ring is denoted as $\mathcal{R}[x_1, \dots, x_n]$. Polynomials with two or three variables are called dyadic/binary, or triadic/ternary, etc. Polynomials with one, two, or three terms are called a monomial, binomial, or trinomial, etc. Degree of a term in a polynomial is the sum of its exponents. Degree of a polynomial is the highest degree of its terms with non-zero coefficients. Homogeneous polynomial or form is a polynomial whose terms have the same degree. Homogeneous polynomials of the first, second, or third degree are called linear, quadratic, or cubic, etc. Quadratic form $q: \mathcal{R}^n \mapsto \mathcal{R}$ over a commutative ring $\mathcal{R}$ is a homogeneous quadratic n-ary polynomial over the ring: $q(x) = \sum_{i=1}^n \sum_{j=i}^n q_{ij} x_i x_j$ where $x_i, q_{ij} \in \mathcal{R}$. Kronecker matrix $A$ of a quadratic form is the symmetric matrix defined by $a_{ij} = q_{ij} + q_{ji}$.

### Field

Division ring (除环) is a ring with multiplicative inverses for all nonzero elements. Division is the inverse operation of multiplication $+$, defined for all nonzero elements of a division ring. Field (域) $(\mathbb{F}, (+, ×))$ is a commutative division ring. Examples of fields: a finite field with four elements; rational numbers $(\mathbb{Q}, (+,×))$; real numbers $(\mathbb{R}, (+,×))$; complex numbers $(\mathbb{C}, (+,×))$.

## Module

Module (模) $(V, (+, \cdot_{\mathcal{R}}))$ over a commutative ring $(\mathcal{R}, (+, ×))$ is a set $V$ endowed with a binary operation $+: V^2 \mapsto V$ and a map $\cdot_{\mathcal{R}}: \mathcal{R} \times V \mapsto V$. Module is a generalization of vector space. Submodule of an $\mathcal{R}$-module is an $\mathcal{R}$-module consisting of a subset and the same module structure. Direct product (直积) $(\prod_{\alpha \in A} V_\alpha, (+, \cdot))$ of an indexed family $(V_\alpha)_{\alpha \in A}$ of $\mathcal{R}$-modules is the $\mathcal{R}$-module consisting of the Cartesian product and addition and scalar multiplication defined as: $(v_\alpha) + (v_\alpha') = (v_\alpha + v_\alpha')$, $c (v_\alpha) = (c v_\alpha)$, where $(v_\alpha)$ is an "A-tuple". External direct sum (直和), or coproduct (余积) in the category of modules, $(\oplus_{\alpha \in A} V_\alpha, (+, \cdot))$ of an indexed family of $\mathcal{R}$-modules is the submodule of their direct product that consists of A-tuples with a finite number of nonzero components. The direct product and the direct sum of a finite family of modules are identical. We say an $\mathcal{R}$-module $W$ is the internal direct sum of an indexed family $(V_\alpha)_{\alpha \in A}$ of its submodules if every element of the module has a unique expression as a finite sum $\sum_\alpha v_\alpha$ of elements of the submodules: $W = \oplus_{\alpha \in A} V_\alpha$.

## Misc

Algebra $(A, (+, \cdot_{\mathbb{R}}, ×))$ over $\mathbb{R}$ is a real vector space $(A, (+, \cdot_{\mathbb{R}}))$ endowed with a bilinear product map $×: A^2 \mapsto A$. Real-valued function algebra $(\mathbb{R}^X, (+, \cdot_{\mathbb{R}}, ×))$ is a real-valued function space endowed with pointwise multiplication. Every polynomial ring over a ring is a commutative free algebra with an identity over the underlying ring, i.e. a free object in the category of commutative algebras.

Unary algebra.

Banach algebra.