Home > Glossary of ring theory

1 Definition of a ring

A ring is an abelian group (R,+) together with an associative operation * which is distributive over + and has an identity element 1 with respect to *. The operation + is referred to as the addition and * is referred to as the multiplication. The identity element with respect to + is written as 0.

The ring with just one element is called the trivial ring.

Characteristic

The characteristic of a ring is the smallest positive integer n satisfying n1=0 if it exists and 0 otherwise. In particular ne=0 for all elements e of the ring.

2 Types of elements

Central

An element r of a ring R is central if xr = rx for all x in R. The set of all central elements forms a subring of R, known as the center of R.

Idempotent

An element e of a ring is idempotent if e² = e.

Irreducible

An element x of a ring is irreducible if for any elements a and b such that x=a b, either a or b is a unit. Note that every irreducible is prime, but not necessarily vice versa.

Prime

An element x of a ring is prime if for any elements a and b such that x=a b, either x divides a or x divides b.

Nilpotent

An element r of R is nilpotent if there exists a positive integer n such that rⁿ = 0.

Unit or invertible element

An element r of the ring R is a unit if there exists an element r^-1 such that rr^-1=r^-1r=1. This element r^-1 is uniquely determined by r and is called the multiplicative inverse of r. The set of units forms a groupIn mathematics, a group is a set, together with a binary operation satisfying certain axioms, detailed below. The branch of mathematics which studies groups is called group theory. The historical origin of group theory goes back to the works of Evariste G under multiplication.

Zero divisorIn abstract algebra, a non-zero element a of a ring R is a left zero divisor if there exists a non-zero b such that ab 0. Right zero divisors are defined analogously. An element that is both a left and a right zero divisor is simply called a zero divisor

A nonzero element r of R is said to be a zero divisor if there exists s ≠ 0 such that sr=0 or rs=0. If a ring has a Zero divisor which is also a unit, then the ring has no other elements and is the trivial ring.

3 Homomorphisms and ideals

Factor ring

Given a ring R and an ideal I of R, the factor ring is the set R/I of cosets {a+I : a∈R} together with operations (a+I)+(b+I)=(a+b)+I and (a+I)*(b+I)=ab+I. The relationship between ideals, homomorphisms, and factor rings is summed up in the fundamental theorem on homomorphismsIn abstract algebra, for a number of algebraic structures, the fundamental theorem on homomorphisms relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism. For groups, the theorem states.

Finitely generated ideal

A left ideal I is finitely generated if there exist finitely many elements a₁,...,a_n such that I = Ra₁ + ... + Ra_n. A right ideal I is finitely generated if there exist finitely many elements a₁,...,a_n such that I = a₁R + ... + a_nR. A two-sided ideal I is finitely generated if there exist finitely many elements a₁,...,a_n such that I = Ra₁R + ... + Ra_nR.

IdealIn ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. For instance, in rings one studies prime ideals instead of prime numbers, one defines coprime ideals as a generalizati

A left ideal I of R is a subgroup or (R,+) such that aI ⊆ I for all a∈R. A right ideal is a subgroup of (R,+) such that Ia⊆I for all a∈R. An ideal (sometimes for emphasis: a two-sided ideal) is a subgroup which is both a left ideal and a right ideal.

Jacobson radicalIn ring theory, a branch of abstract algebra, the Jacobson radical of a ring R is an ideal of R which contains those elements of R which in a sense are "close to zero". It is denoted by J R and can be defined in the following equivalent ways: the intersec

The intersection of all maximal left ideals in a ring forms a two-sided ideal, the Jacobson radical of the ring.

KernelAlgebra In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. The definition of kernel takes various forms in various conte of a ring homomorphism

It is the preimage of 0 in the codomain of a ring homomorphism. Every ideal is the kernel of a ring homomorphism and vice versa.

Maximal ideal

A left ideal of the ring R which is not contained in any other left ideal but R itself is called a maximal left ideal. Maximal right ideals are defined similarly. In commutative rings, there is no difference, and one speaks simply of maximal ideals.

Nilradical

The set of all nilpotent elements in a commutative ring forms an ideal, the nilradical of the ring. The nilradical is equal to the intersection of all the Prime Ideals. It is 'not' equal, in general, to the Jacobson Radical.

Prime ideal

An ideal P in a commutative ring R is prime if P ≠ R and if for all a and b in R with ab in P, we have a in P or b in P. Every maximal ideal in a commutative ring is prime. There is also a definition of prime ideal for noncommutative rings.

Principal ideal

a principal left ideal in the ring R is a left ideal of the form Ra for some element a of R; a principal right ideal is a right ideal of the form aR for some element a of R; a principal ideal is a two-sided ideal of the form RaR for some element a of R.

Radical of an ideal

The radical of an ideal I in a commutative ring consists of all those ring elements a power of which lies in I. It is equal to the intersection of all maximal ideals containing I.

Ring homomorphism

A function f : R → S between rings (R,+,*) and (S,⊕,×) is a ring homomorphism if it has the special properties that

f(a + b) = f(a) ⊕ f(b)

f(a * b) = f(a) × f(b)

f(1) = 1

for any elements a and b of R.

Ring monomorphism

A ring homomorphism that is injective is a ring monomorphism.

Ring epimorphism

A ring homomorphism that is surjective is a ring epimorphism.

Ring isomorphism

A ring homomorphism that is bijective is a ring isomorphism. The inverse of an isomorphism, it turns out, is also a ring isomorphism. Two rings are isomorphic if there exists a ring isomorphism between them. Isomorphic rings can be thought as essentially the same, only with different labels on the individual elements.