A collection of objects and morphisms between them

A set of elements and operations on those elements

A group of transformations on a set

A ring with a unit

A map between two categories that preserves the structure of the categories

A function between two sets that preserves the operations on the sets

A group homomorphism between two groups

A ring homomorphism between two rings

A map between two functors that preserves the structure of the functors

A function between two functions that preserves the operations on the functions

A group homomorphism between two group homomorphisms

A ring homomorphism between two ring homomorphisms

A homomorphism from the group to the group of invertible linear transformations of a vector space

A function from the group to the set of all linear transformations of a vector space

A group homomorphism from the group to the group of units of a ring

A ring homomorphism from the group to the group of invertible elements of a field

A function from the group to the complex numbers that is constant on conjugacy classes

A function from the group to the real numbers that is constant on conjugacy classes

A group homomorphism from the group to the group of units of a ring

A ring homomorphism from the group to the group of invertible elements of a field

A set of equations that relate the characters of a group

A set of equations that relate the representations of a group

A set of equations that relate the elements of a group

A set of equations that relate the morphisms of a category

A theorem that states that every finite-dimensional representation of a group is completely reducible

A theorem that states that every finite-dimensional representation of a group is semisimple

A theorem that states that every finite-dimensional representation of a group is indecomposable

A theorem that states that every finite-dimensional representation of a group is simple

A theorem that states that every finite division ring is a field

A theorem that states that every finite-dimensional division algebra is a field

A theorem that states that every finite-dimensional algebra is semisimple

A theorem that states that every finite-dimensional algebra is simple

The largest nilpotent ideal of a ring

The largest radical ideal of a ring

The largest prime ideal of a ring

The largest maximal ideal of a ring

A theorem that states that every semisimple ring is a direct product of simple rings

A theorem that states that every semisimple algebra is a direct product of simple algebras

A theorem that states that every finite-dimensional semisimple algebra is a direct product of simple algebras

A theorem that states that every finite-dimensional semisimple ring is a direct product of simple rings

A theorem that states that every finite-dimensional division algebra over a field is a central simple algebra

A theorem that states that every finite-dimensional central simple algebra over a field is a division algebra

A theorem that states that every finite-dimensional algebra over a field is a central simple algebra

A theorem that states that every finite-dimensional central simple algebra over a field is an algebra

A theorem that states that every finitely generated module over a principal ideal domain is a direct sum of cyclic modules

A theorem that states that every finitely generated module over a Euclidean domain is a direct sum of cyclic modules

A theorem that states that every finitely generated module over a Dedekind domain is a direct sum of cyclic modules

A theorem that states that every finitely generated module over a field is a direct sum of cyclic modules

A theorem that states that every ideal in a Noetherian ring is finitely generated

A theorem that states that every ideal in a Dedekind domain is finitely generated

A theorem that states that every ideal in a principal ideal domain is finitely generated

A theorem that states that every ideal in a Euclidean domain is finitely generated

A theorem that states that every finitely generated module over a Noetherian ring is a direct sum of indecomposable modules

A theorem that states that every finitely generated module over a Dedekind domain is a direct sum of indecomposable modules

A theorem that states that every finitely generated module over a principal ideal domain is a direct sum of indecomposable modules

A theorem that states that every finitely generated module over a Euclidean domain is a direct sum of indecomposable modules

A theorem that states that every semisimple ring is a direct product of simple rings

A theorem that states that every semisimple algebra is a direct product of simple algebras

A theorem that states that every finite-dimensional semisimple algebra is a direct product of simple algebras

A theorem that states that every finite-dimensional semisimple ring is a direct product of simple rings