Commutative algebra | StudentB Wikipedia Search

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Algebraic structure → Ring theory Ring theory
Basic concepts Rings • 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 $\mathbb {Z}$ • Terminal ring $0=\mathbb {Z} /1\mathbb {Z}$ Related structures • Field • Finite field • Non-associative ring • Lie ring • Jordan ring • Semiring • Semifield
Commutative algebra Commutative rings • 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 $\mathbb {Z} _{p}$ • p-adic numbers $\mathbb {Q} _{p}$ • Prüfer p-ring $\mathbb {Z} (p^{\infty })$
Noncommutative algebra Noncommutative rings • Division ring • Semiprimitive ring • Simple ring • Commutator Noncommutative algebraic geometry Free algebra Clifford algebra • Geometric algebra Operator algebra
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Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers $\mathbb {Z}$ ; and p-adic integers.^[1]

Commutative algebra is the main technical tool of algebraic geometry, and many results and concepts of commutative algebra are strongly related with geometrical concepts.

The study of rings that are not necessarily commutative is known as noncommutative algebra; it includes ring theory, representation theory, and the theory of Banach algebras.

^ Atiyah and Macdonald, 1969, Chapter 1