# Metamath Proof Explorer

## Table of Contents - 10. BASIC ALGEBRAIC STRUCTURES

- Monoids
- Magmas
- Identity elements
- Iterated sums in a magma
- Semigroups
- Definition and basic properties of monoids
- Monoid homomorphisms and submonoids
- Iterated sums in a monoid
- Free monoids
- Examples and counterexamples for magmas, semigroups and monoids

- Groups
- Definition and basic properties
- Group multiple operation
- Subgroups and Quotient groups
- Cyclic monoids and groups
- Elementary theory of group homomorphisms
- Isomorphisms of groups
- Group actions
- Centralizers and centers
- The opposite group
- Symmetric groups
- p-Groups and Sylow groups; Sylow's theorems
- Direct products
- Free groups
- Abelian groups
- Simple groups

- Rings
- Multiplicative Group
- Ring unit
- Definition and basic properties of unital rings
- Opposite ring
- Divisibility
- Ring primes
- Ring homomorphisms

- Division rings and fields
- Definition and basic properties
- Subrings of a ring
- Absolute value (abstract algebra)
- Star rings

- Left modules
- Definition and basic properties
- Subspaces and spans in a left module
- Homomorphisms and isomorphisms of left modules
- Subspace sum; bases for a left module

- Vector spaces
- Definition and basic properties

- Ideals
- The subring algebra; ideals
- Two-sided ideals and quotient rings
- Principal ideal rings. Divisibility in the integers
- Nonzero rings and zero rings
- Left regular elements. More kinds of rings

- The complex numbers as an algebraic extensible structure
- Definition and basic properties
- Ring of integers
- Algebraic constructions based on the complex numbers
- Signs as subgroup of the complex numbers
- Embedding of permutation signs into a ring
- The ordered field of real numbers

- Generalized pre-Hilbert and Hilbert spaces
- Definition and basic properties
- Orthocomplements and closed subspaces
- Orthogonal projection and orthonormal bases