Basic Language
Algebraic structures are defined primarily as sets with operations.
- Algebraic structure
- Subobjects: subgroup, subring, subalgebra, submodule etc.
- Binary operation
- Closure of an operation
- Associative property
- Distributive property
- Commutative property
- Unary operator
- Additive inverse, multiplicative inverse, inverse element
- Identity element
- Cancellation property
- Additive inverse, multiplicative inverse, inverse element
- Finitary operation
- Arity
Structure preserving maps called homomorphisms are vital in the study of algebraic objects.
- Homomorphisms
- Kernels and cokernels
- Image and coimage
- Epimorphisms and monomorphisms
- Isomorphisms
- Isomorphism theorems
- Isomorphisms
There are several basic ways to combine algebraic objects of the same type to produce a third object of the same type. These constructions are used throughout algebra.
- Direct sum
- Direct limit
- Direct product
- Inverse limit
- Quotient objects: quotient group, quotient ring, quotient module etc.
- Tensor product
Advanced concepts:
- Category theory
- Category of groups
- Category of abelian groups
- Category of rings
- Category of modules
- Morita equivalence, Morita duality
- Category of vector spaces
- Category of groups
- Homological algebra
- Filtration (algebra)
- Exact sequence
- Functor
- Zorn's lemma
Read more about this topic: List Of Abstract Algebra Topics
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