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The ring of integers O K is a finitely-generated Z-module.Indeed, it is a free Z-module, and thus has an integral basis, that is a basis b 1, ..., b n ∈ O K of the Q-vector space K such that each element x in O K can be uniquely represented as
Using this and denoting repeated addition by a multiplication by a positive integer allows identifying abelian groups with modules over the ring of integers. Any ring homomorphism induces a structure of a module: if f : R → S is a ring homomorphism, then S is a left module over R by the multiplication: rs = f(r)s. If R is commutative or if f ...
The ring of integers of Q( √ −19 ), consisting of the numbers a + b √ −19 / 2 where a and b are integers and both even or both odd. It is a principal ideal domain that is not Euclidean.This was proved by Theodore Motzkin and was the first case known.
Similarly, a polynomial ring with integer coefficients is the free commutative ring over its set of variables, since commutative rings and commutative algebras over the integers are the same thing. Graded structure
The ring of algebraic integers is a Bézout domain, as a consequence of the principal ideal theorem. If the monic polynomial associated with an algebraic integer has constant term 1 or −1, then the reciprocal of that algebraic integer is also an algebraic integer, and each is a unit , an element of the group of units of the ring of algebraic ...
Integers in the same congruence class a ≡ b (mod n) satisfy gcd(a, n) ... The multiplicative group of integers modulo n, which is the group of units in this ring, ...
An important property of the ring of integers is that it satisfies the fundamental theorem of arithmetic, that every (positive) integer has a factorization into a product of prime numbers, and this factorization is unique up to the ordering of the factors. This may no longer be true in the ring of integers O of an algebraic number field K.
The integers form a ring which is the most basic one, in the following sense: for any ring, there is a unique ring homomorphism from the integers into this ring. This universal property , namely to be an initial object in the category of rings , characterizes the ring Z {\displaystyle \mathbb {Z} } .