Glossary of Ring Theory - Homomorphisms and Ideals

Homomorphisms and Ideals

Factor ring or quotient ring

Given a ring R and an ideal I of R, the factor ring is the ring formed by the set R/I of cosets {a+I : a∈R} together with the 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 homomorphisms.

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.

Ideal

A left ideal I of R is a subgroup of 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 called a two-sided ideal for emphasis) is a subgroup which is both a left ideal and a right ideal.

Jacobson radical

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

Kernel of a ring homomorphism

The kernel of a ring homomorphism f : R → S is the set of all elements x of R such that f(x) = 0. Every ideal is the kernel of a ring homomorphism and vice versa.

Maximal ideal

A left ideal M of the ring R is a maximal left ideal if M ≠ R and the only left ideals containing M are R and M itself. Maximal right ideals are defined similarly. In commutative rings, there is no difference, and one speaks simply of maximal ideals.

Nil ideal

An ideal is nil if it consists only of nilpotent elements.

Nilpotent ideal

An ideal I is nilpotent if the power Ik is {0} for some positive integer k. Every nilpotent ideal is nil, but the converse is not true in general.

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 ring's prime ideals. It is contained in, but in general not equal to, the ring's 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 a 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 prime ideals containing I.

Ring homomorphism

A function f : R → S between rings (R,+,*) and (S,⊕,×) is a ring homomorphism if it satisfies

f(a + b) = f(a) ⊕ f(b)

f(a * b) = f(a) × f(b)

f(1) = 1

for all elements a and b of R.

Ring monomorphism

A ring homomorphism that is injective is a ring monomorphism.

Ring isomorphism

A ring homomorphism that is bijective is a ring isomorphism. The inverse of a ring isomorphism 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.

Trivial ideal

Every nonzero ring R is guaranteed to have two ideals: the zero ideal and the entire ring R. These ideals are usually referred to as trivial ideals. Right ideals, left ideals, and two-sided ideals other than these are called nontrivial.