ring of integers

On the other hand, the ring of integers in a number field is always a Dedekind domain.

In fact, every ideal of the ring of integers is principal.

By contrast, the ring of integers is not infinitely divisible.

Therefore, the ring of integers of F is an integral domain.

The rings of integers in number fields are Dedekind domains.

Abelian groups are modules over the ring of integers.

Generalizing this simple result to more general rings of integers is a basic problem in algebraic number theory.

The ring of integers is semiprimitive, but not semisimple.

The ring of integers Z has dimension 1.

The ring of integers Z is a reduced ring.

Let K be an algebraic number field with R the ring of integers.

The problem for the ring of integers of algebraic number fields other than those covered by the results above remains open.

Higher K groups can also be employed and interpreted arithmetically in connection to rings of integers.

The motivating example is the ring of integers with the two operations of addition and multiplication.

There is a natural way to make it a ring by adding negative numbers to the set, thus obtaining the ring of integers.

Each algebraic integer belongs to the ring of integers of some number field.

The ring of integers Z forms an initial object in Ring.

The ring of integers of a number field is the unique maximal order in the field.

Some number fields have rings of integers that do not form a unique factorization domain, for example the field .

That is, the ring of integers of the field has unique factorization for .

Equivalently, its ring of integers has unique factorization.

The leading example is the case where A is a number field K and is its ring of integers.

It then encodes the ramification data for prime ideals of the ring of integers.

Consider the ring of integers Z and the ideal of even numbers, denoted by 2Z.

That is, every abelian group is a module over the ring of integers Z in a unique way.