ring theory

ring theory is a branch of abstract algebra that deals with the study of rings and their properties, which are algebraic structures consisting of a set together with two binary operations, usually called addition and multiplication, that satisfy certain properties. The development of ring theory is closely tied to the work of David Hilbert, Emmy Noether, and Richard Dedekind, who made significant contributions to the field of algebraic number theory and the study of ideals in commutative rings. The study of rings has numerous applications in physics, computer science, and cryptography, particularly in the work of Claude Shannon and Alan Turing.

Introduction to Ring Theory

The study of ring theory began with the work of Richard Dedekind on ideal theory in the late 19th century, which was later developed by David Hilbert and Emmy Noether in the early 20th century. The concept of a ring was formally introduced by David Hilbert in his work on invariant theory, and was later developed by Emmy Noether in her work on abstract algebra. The development of ring theory was also influenced by the work of André Weil on algebraic geometry and the study of algebraic curves by Bernhard Riemann and Felix Klein. The study of rings has numerous applications in number theory, particularly in the work of Andrew Wiles on Fermat's Last Theorem and the study of elliptic curves by Gerd Faltings and Andrew Wiles.

Definitions and Examples

A ring is a mathematical structure consisting of a set together with two binary operations, usually called addition and multiplication, that satisfy certain properties, such as associativity and distributivity. Examples of rings include the integers, the rational numbers, and the real numbers, as well as the Gaussian integers and the Eisenstein integers, which were studied by Carl Friedrich Gauss and Ferdinand Eisenstein. Other examples of rings include the matrix rings and the group rings, which were studied by Arthur Cayley and William Rowan Hamilton. The study of rings is closely tied to the study of fields, particularly in the work of Évariste Galois and Niels Henrik Abel on Galois theory.

Types of Rings

There are several types of rings, including commutative rings, non-commutative rings, and division rings, which were studied by Joseph Wedderburn and Emmy Noether. Examples of commutative rings include the integers and the polynomial rings, which were studied by David Hilbert and Emmy Noether. Examples of non-commutative rings include the matrix rings and the quaternion rings, which were studied by William Rowan Hamilton and Arthur Cayley. The study of rings is also closely tied to the study of algebras, particularly in the work of Hermann Grassmann and Elie Cartan on exterior algebra and differential forms.

Ring Homomorphisms and Ideals

A ring homomorphism is a function between two rings that preserves the operations of addition and multiplication, and was studied by David Hilbert and Emmy Noether. The kernel of a ring homomorphism is an ideal of the domain ring, and was studied by Richard Dedekind and Emmy Noether. Ideals are important in the study of rings, particularly in the work of David Hilbert and Emmy Noether on ideal theory. The study of ideals is closely tied to the study of modules, particularly in the work of Emmy Noether and Richard Brauer on module theory. The study of ring homomorphisms and ideals has numerous applications in algebraic geometry, particularly in the work of André Weil and Alexander Grothendieck.

Ring Constructions and Extensions

There are several ways to construct new rings from existing ones, including the direct product of rings and the quotient ring of a ring by an ideal, which were studied by David Hilbert and Emmy Noether. The study of ring extensions is closely tied to the study of field extensions, particularly in the work of Évariste Galois and Niels Henrik Abel on Galois theory. The study of ring constructions and extensions has numerous applications in number theory, particularly in the work of Andrew Wiles on Fermat's Last Theorem and the study of elliptic curves by Gerd Faltings and Andrew Wiles. The study of ring theory is also closely tied to the study of representation theory, particularly in the work of Richard Brauer and Emmy Noether on module theory and the study of groups by William Burnside and John Conway. Category:Abstract algebra