Algebraic structures: monoids, groups and rings

前置知识

11 项必需的前置知识（一直追溯到底）

你将学会

• Define monoids, groups, rings and fields, and test examples such as (ℕ, +), (ℤ, +), (ℤ, ×), ℤ/nℤ and 2 × 2 matrices.
• Prove that identities and inverses are unique when they exist.
• Explain why ℤ is a ring but not a field, while ℚ is a field.
• Recognise structure-preserving maps (homomorphisms) between these structures.

定义

• A binary operation on a set S is a function S × S → S.
• A monoid is a set with an associative binary operation and an identity element; a group is a monoid in which every element has an inverse.
• A ring is a set with operations + and × such that (R, +) is an abelian group, (R, ×) is a monoid, and × distributes over +. A field is a commutative ring with 0 ≠ 1 in which every non-zero element has a multiplicative inverse.
• A homomorphism is a function between structures of the same kind that preserves the operations and identity elements.

定理

• In a monoid the identity element is unique, and each element has at most one inverse.
• (ℕ, +) is a monoid but not a group, while (ℤ, +) is a group.
• ℤ/nℤ is a field if and only if n is prime.
• The functions from a set to itself form a monoid under composition, and the bijections form a group.

常见误解

• Thinking an identity element must be 0 or 1; for composition it is the identity function.
• Assuming every operation is commutative; matrix multiplication and composition of functions are not.
• Forgetting closure: the odd integers are not closed under addition.

通向

可选发现

• Functors A monoid can be viewed as a category with one object, and a monoid homomorphism is then exactly a functor between such categories.
• Monads A monad is a monoid in a category of endofunctors, with composition as the product: the monoid idea reappears one level up.探奇小径 →
• ← Integers
• ← Rational numbers
• ← Composition
• ← Orders, lattices and well-orders

学习路径

• Accelerated secondary progression
• Undergraduate mathematics major
• Computer science and formal methods

延伸阅读

• Michael Artin, Algebra (1991)
• Joseph A. Gallian, Contemporary Abstract Algebra