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.

**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** (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** 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 $a_{ij} = q_{ij} + q_{ji}$.

**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** (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:
$W = \oplus_{\alpha \in A} V_\alpha$.

**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.