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Ring theory - Elementary introduction

Ring theory - Elementary introduction: Encyclopedia II - Ring theory - Elementary introduction

Ring theory - Definition. Formally, a ring is an abelian group (R, +), together with a second binary operation * such that for all a, b and c in R, a * (b * c) = (a * b) * c a * (b + c) = (a * b) + (a * c) (a + b) * c = ( ...

Ring theory: Encyclopedia II - Ring theory - Elementary introduction

Ring theory - Elementary introduction

Ring theory - Definition

Formally, a ring is an abelian group (R, +), together with a second binary operation * such that for all a, b and c in R,

a * (b * c) = (a * b) * c a * (b + c) = (a * b) + (a * c) (a + b) * c = (a * c) + (b * c)

and such that there exists a multiplicative identity, or unity, that is, an element 1 so that for all a in R,

a * 1 = 1 * a = a

It is simple to show that any ring in which 1 = 0 must have just one element; any such ring is called a zero ring.

Rings that sit inside other rings are called subrings. Maps between rings which respect the ring operations are called ring homomorphisms. Rings, together with ring homomorphisms, form a category. Closely related is the notion of ideals, certain subsets of rings which arise as kernels of homomorphisms and can serve to define factor rings. Basic facts about ideals, homomorphisms and factor rings are recorded in the isomorphism theorems and in the Chinese remainder theorem.

A ring is called commutative if its multiplication is commutative. Commutative rings resemble familiar number systems, and various definitions for commutative rings are designed to recover properties known from the integers. Commutative rings are also important in algebraic geometry. In commutative ring theory, numbers are often replaced by ideals, and the definition of prime ideal tries to capture the essence of prime numbers. Integral domains, non-trivial commutative rings where no two non-zero elements multiply to give zero, generalize another property of the integers and serve as the proper realm to study divisibility. Principal ideal domains are integral domains in which every ideal can be generated by a single element, another property shared by the integers. Euclidean domains are integral domains in which the Euclidean algorithm can be carried out. Important examples of commutative rings can be constructed as rings of polynomials and their factor rings. Summary: Euclidean domain => principal ideal domain => unique factorization domain => integral domain => Commutative ring.

Non-commutative rings resemble rings of matrices in many respects. Following the model of algebraic geometry, attempts have been made recently at defining non-commutative geometry based on non-commutative rings. Non-commutative rings and associative algebras (rings that are also vector spaces) are often studied via their categories of modules. A module over a ring is an abelian group that the ring acts on as a ring of endomorphisms, very much akin to the way fields (integral domains in which every non-zero element is invertible) act on vector spaces. Examples of non-commutative rings are given by rings of square matrices or more generally by rings of endomorphisms of abelian groups or modules, and by monoid rings.