Transcendental number theory

• Subrings

• Ideal

• Quotient ring

• Fractional ideal

• Total ring of fractions

• Product of rings

• Free product of associative algebras

• Tensor product of algebras

Ring homomorphisms

• Kernel

• Inner automorphism

• Frobenius endomorphism

Algebraic structures

• Module

• Associative algebra

• Graded ring

• Involutive ring

• Category of rings

• Initial ring ${\displaystyle \mathbb {Z} }$

• Terminal ring ${\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }$

Related structures

• Field

• Finite field

• Non-associative ring

• Lie ring

• Jordan ring

• Semiring

• Semifield

• Integral domain

• Integrally closed domain

• GCD domain

• Unique factorization domain

• Principal ideal domain

• Euclidean domain

• Field

• Finite field

• Polynomial ring

• Formal power series ring

Algebraic number theory

• Algebraic number field

• Integers modulo n

• Ring of integers

• p-adic integers ${\displaystyle \mathbb {Z} _{p}}$

• p-adic numbers ${\displaystyle \mathbb {Q} _{p}}$

• Prüfer p-ring ${\displaystyle \mathbb {Z} (p^{\infty })}$

• Division ring

• Semiprimitive ring

• Simple ring

• Commutator

Noncommutative algebraic geometry

Free algebra

Clifford algebra

• Geometric algebra