Glossary of Ring Theory - Types of Rings

Types of Rings

Abelian ring

A ring in which all idempotents are central is called an Abelian ring. Such rings need not be commutative.

Artinian ring

A ring satisfying the descending chain condition for left ideals is left artinian; if it satisfies the descending chain condition for right ideals, it is right artinian; if it is both left and right artinian, it is called artinian. Artinian rings are noetherian.

Boolean ring

A ring in which every element is multiplicatively idempotent is a boolean ring.

Commutative ring

A ring R is commutative if the multiplication is commutative, i.e. rs=sr for all r,s∈R.

Dedekind domain

A Dedekind domain is an integral domain in which every ideal has a unique factorization into prime ideals.

Division ring or skew field

A ring in which every nonzero element is a unit and 1≠0 is a division ring.

Domain (ring theory)

A domain is a ring without zero divisors and in which 1≠0. This is the noncommutative generalization of integral domain.

Euclidean domain

A Euclidean domain is an integral domain in which a degree function is defined so that "division with remainder" can be carried out. It is so named because the Euclidean algorithm is a well-defined algorithm in these rings. All Euclidean domains are principal ideal domains.

Field

A field is a commutative division ring. Every finite division ring is a field, as is every finite integral domain.

Finitely presented algebra

If R is a commutative ring and A is an R-algebra, then A is a finitely presented R-algebra if it is a quotient of a polynomial ring over R in finitely many variables by a finitely generated ideal.

Hereditary ring

A ring is left hereditary if its left ideals are all projective modules. Right hereditary rings are defined analogously.

Integral domain or entire ring

A commutative ring without zero divisors and in which 1≠0 is an integral domain.

Invariant basis number

A ring R has invariant basis number if Rm isomorphic to Rn as R-modules implies m=n.

Local ring

A ring with a unique maximal left ideal is a local ring. These rings also have a unique maximal right ideal, and the left and the right unique maximal ideals coincide. Certain commutative rings can be embedded in local rings via localization at a prime ideal.

Noetherian ring

A ring satisfying the ascending chain condition for left ideals is left noetherian; a ring satisfying the ascending chain condition for right ideals is right noetherian; a ring that is both left and right noetherian is noetherian. A ring is left noetherian if and only if all its left ideals are finitely generated; analogously for right noetherian rings.

Perfect ring

A left perfect ring is one satisfying the descending chain condition on right principal ideals. They are also characterized as rings whose flat left modules are all projective modules. Right perfect rings are defined analogously. Artinian rings are perfect.

Prime ring

A non-trivial ring R is called a prime ring if for any two elements a and b of R with aRb = 0, we have either a = 0 or b = 0. This is equivalent to saying that the zero ideal is a prime ideal. Every simple ring and every domain is a prime ring.

Primitive ring

A left primitive ring is a ring that has a faithful simple left R-module. Every simple ring is primitive. Primitive rings are prime.

Principal ideal domain

An integral domain in which every ideal is principal is a principal ideal domain. All principal ideal domains are unique factorization domains.

Quasi-Frobenius ring

a special type of Artinian ring which is also a self-injective ring on both sides. Every semisimple ring is quasi-Frobenius.

Self-injective ring

A ring R is left self-injective if the module _RR is an injective module. While rings with unity are always projective as modules, they are not always injective as modules.

Semiprimitive ring or Jacobson semisimple ring

This is a ring whose Jacobson radical is zero. Von Neumann regular rings and primitive rings are semiprimitive, however quasi-Frobenius rings and local rings are usually not semiprimitive.

Semisimple ring

A semisimple ring is a ring R that has a "nice" decomposition, in the sense that R is a semisimple left R-module. Every semisimple ring is also Artinian, and has no nilpotent ideals. The Artin–Wedderburn theorem asserts that every semisimple ring is a finite product of full matrix rings over division rings.

Simple ring

A non-zero ring which only has trival two-sided ideals (the zero ideal, the ring itself, and no more) is a simple ring.

Trivial ring or zero ring

The ring consisting only of a single element 0=1. Zero ring also has another meaning, see below.

Unique factorization domain or factorial ring

An integral domain R in which every non-zero non-unit element can be written as a product of prime elements of R. This essentially means that every non-zero non-unit can be written uniquely as a product of irreducible elements.

von Neumann regular ring

A ring for which each element a can be expressed as a=axa for another element x in the ring. Semisimple rings are von Neumann regular.

Zero ring or null ring

A rng (ring without 1) in which the product of any two elements is 0 (the additive neutral element). Also, the zero ring, the ring consisting only of a single element 0=1 (see trivial ring, above).