Abstract algebra studies the properties of mathematical structures with operators through the lens of abstraction. Although it is not at all a complex topic per se, since it often extends familiar topics, it is one with many definitions and notions to keep track of. Therefore, here is a list of some key concepts in abstract algebra.

Wikipedia has a good overview of its history on the Abstract Algebra page and a list of Basic Concepts on its own page.

1. Basic Set-Theoretic Structures

1.1 Set

  • A collection of distinct elements with no additional structure beyond membership.

1.2 Binary Operation

  • A rule that combines two elements of a set to produce another element of the same set. For example, addition and multiplication are binary operations on numbers.

2. Algebraic Structures

2.1 Group

  • A set $ G $ with a binary operation satisfying:
    1. Closure: $ a \cdot b \in G $ for all $ a, b \in G $.
    2. Associativity: $ (a \cdot b) \cdot c = a \cdot (b \cdot c) $.
    3. Identity: There exists $ e \in G $ such that $ e \cdot a = a \cdot e = a $ for all $ a \in G $.
    4. Inverse: For each $ a \in G $, there is an inverse $ a^{-1} \in G $ such that $ a \cdot a^{-1} = e $.

2.2 Abelian Group

  • A group where the binary operation is commutative, i.e., $ a \cdot b = b \cdot a $ for all $ a, b \in G $.

2.3 Ring

  • A set $ R $ with two binary operations (addition and multiplication) where:
    1. Addition forms an abelian group.
    2. Multiplication is associative.
    3. Distributivity: Multiplication distributes over addition.

2.4 Field

  • A commutative ring where every nonzero element has a multiplicative inverse. Fields allow division by nonzero elements, making them more structured than rings.

2.5 Module

  • A generalization of vector spaces where scalars come from a ring (instead of a field). A module has a scalar multiplication that satisfies similar axioms to those of vector spaces.

2.6 Vector Space

  • A special case of a module where scalars come from a field. A vector space consists of vectors that can be added and scaled by elements from the field, satisfying specific axioms.

2.7 Algebra

  • A vector space (or module) with a bilinear multiplication operation that combines vectors to form another vector. Examples include matrix algebras and polynomial algebras.

2.8 Monoid

  • A set $ M $ with a binary operation that is associative and has an identity element, but inverses are not required (unlike groups).

2.9 Semigroup

  • A set $ S $ with an associative binary operation. Semigroups do not require an identity element or inverses.

3. Morphisms (Structure-Preserving Maps)

3.1 Homomorphism

  • A function between two algebraic structures (such as groups, rings, or vector spaces) that preserves the relevant operations. For example:
    • Group Homomorphism: $ \phi(a \cdot b) = \phi(a) \cdot \phi(b) $ for all $ a, b \in G $.
    • Ring Homomorphism: $ \phi(a + b) = \phi(a) + \phi(b) $ and $ \phi(a \cdot b) = \phi(a) \cdot \phi(b) $.

3.2 Isomorphism

  • A bijective homomorphism between two algebraic structures. If $ \phi: A \to B $ is an isomorphism, the structures $ A $ and $ B $ are considered algebraically identical.

3.3 Automorphism

  • A special case of an isomorphism where the domain and codomain are the same, i.e., $ \phi: A \to A $. It maps a structure back to itself while preserving its operations.

4. Generalizations and Specializations

4.1 Ideal

  • A subset of a ring $ R $ that is closed under addition and is stable under multiplication by elements of $ R $. Ideals are used to create quotient rings.

4.2 Quotient Group / Ring / Module

  • The result of partitioning a group, ring, or module by a normal subgroup, ideal, or submodule, respectively. This quotient structure allows the construction of new algebraic structures.

4.3 Simple Group

  • A group that has no nontrivial normal subgroups other than itself and the identity element. Simple groups are building blocks for more complex groups in group theory.

4.4 Tensor Product

  • A construction that combines two vector spaces, modules, or algebras into a new structure. The tensor product plays a key role in multilinear algebra and other areas.

5. Non-Associative Structures

5.1 Lie Algebra

  • A vector space equipped with a non-associative operation called the Lie bracket. This structure is used to study symmetries and is widely applied in physics and geometry. It satisfies:
    1. Bilinearity.
    2. Antisymmetry: $ [a, b] = -[b, a] $.
    3. Jacobi identity.

6. Order-Theoretic and Categorical Structures

6.1 Lattice

  • A partially ordered set (poset) in which any two elements have a least upper bound (join) and a greatest lower bound (meet). Lattices are used in order theory and have applications in logic and algebra.

6.2 Category

  • A higher abstraction that consists of objects and morphisms (arrows) between those objects. Morphisms can be composed in an associative way, and categories generalize many algebraic structures by focusing on relationships between objects.