ringtheory

Ring theory is a branch of abstract algebra that studies rings, algebraic structures consisting of a set equipped with two binary operations: addition and multiplication. In a ring, the additive structure forms an abelian group, multiplication is associative, and multiplication distributes over addition. Many texts also assume a multiplicative identity, making the ring unital; rings without an identity are studied as well. Rings can be commutative or noncommutative, and rings with additional properties are studied in greater depth.

Key concepts in ring theory include ideals, quotient rings, and ring homomorphisms. An ideal is a special subset that allows the construction of quotient rings, which in turn reflect the structure of the original ring. Homomorphisms preserve ring operations and give a way to relate different rings. Prime and maximal ideals play central roles: prime ideals correspond to irreducible elements in a geometric sense, while maximal ideals yield simple quotients. The spectrum of a ring, denoted Spec(R), is a geometric object underpinning much of algebraic geometry.

Common classes of rings include commutative rings, integral domains (rings with no zero divisors), and fields (rings in which every nonzero element is invertible). Special types such as principal ideal domains (PIDs), unique factorization domains (UFDs), Euclidean domains, Noetherian rings, and Artinian rings organize many structural results. For example, Z is a principal ideal domain and a UFD, while polynomial rings over a field serve as a primary and versatile source of examples.

Ring theory supplies foundational tools for many areas of mathematics, including algebraic geometry, number theory, representation theory, and algebraic topology, and it provides a framework for studying algebras and modules over rings.