Divisibility in discrete domains

1. Divisibility in cancellation monoids

HEGL Illustrating Mathematics Seminar

Winter Semester 2024-2025

by Sarah Gramlich

Cancellation monoid

A commutative monoid is called a cancellation monoid if . So there are no different and with .

Example: If is a discrete integral domain, then the set of nonzero elements of forms a discrete cancellation monoid.

Divisibility

If and are elements of a cancellation monoid , then we say that divides , and write , if there exists such that .

Associates

For , and are called associates, written , if each divides the other.

GCD greatest common divisor

is a GCD() if and and for each :

and then

relatively prime: GCD()=1

GCD-monoid/-domain

-GCD-monoid: a cancellation monoid where each pair of elements has a GCD

-GCD-domain: a discrete domain where nonzero elements form a GCD-monoid

Theorem 1.1

, is GCD-monoid

Then:

i) GCD(GCD()= GCD(,GCD())

ii) ×GCD()=GCD()

iii) if =GCD() then GCD()=GCD()

iv) if and GCD()=1 then

Irreducible

A nonunit of a cancellation monoid is irreducible if whenever , then either or is a unit.

Prime

A nonunit is prime if whenever , then or .

Lemma 1

Each irreducible element in a GCD-monoid is prime.

Bounded

cancellation monoid

• is bounded by n, if with , then is a unit for some
• is bounded, if it is bounded by n for some
• is bounded if all are bounded
• a discrete domain is bounded if the nonzero elements form a bounded monoid

Principal ideal

A principal ideal of a commutative monoid is a subset of such that { } for some in

Lemma 2

is a GCD-monoid, satisfying the divisor chain condition and are pairwise relatively prime elements of .
If we can construct :

i)

ii) j then exists :

iii) and are relatively prime for j= 1, 2,...,

Quasi-factorization

are elements of a GCD-monoid satisfying the divisor chain condition, then a family of pairwise relatively prime elements of , such that each is associate of a product of elements of .

Divisibility in discrete domains

2. UFD's and Bézout domains

HEGL Illustrating Mathematics Seminar
Winter Semester 2024-2025

Heide Frank
Heidelberg University

Definition 2.1

a) unique factorization domain (UFD)

• Discrete domain
• Each 0 is a unit or has an essentially unique factorization into irreducible elements:

                and we can reindex so that ~

b) We say that R is factorial if is a UFD

Remark

• A UFD has recognizable units ( is decidable)
• is UFD is factorial

Examples

• discrete fields

Theorem 2.2

Let be a discret domain.
Then:

1. If is factorial, then is a UFD
2. If is a UFD, then is a bounded GCD-domain
3. If is a bounded GCD-domain, then is a quasi-UFD

In other words:
is factorial is UFD is bounded GCD-domain is quasi-UFD

Brouwerian Counterexample 2.3

Let be a binary sequence, the field of rational numbers, , and
Then :

• is a discrete field is UFD

• However, we cannot factor into irreducibles over

is UFD is factorial

Definition 2.4

A multiplicative submonoid of a commutative ring is called saturated if

Theorem 2.5

Let be a discrete domain and a multiplicative submonoid of .
Then:

1. If is a GCD-domain, then so is
2. If saturated and detachable, then has recognizable units

possible project: proof of 1.

Theorem 2.6

Let be a UFD and let be a saturated detachable multiplicative submonoid of .
Then is also a UFD.

Corollary 2.7

If is a UFD and is a detachable saturated multiplicative submonoid of .
Then is also a UFD.

possible project?

Definition 2.8

• A Bézout domain is a discrete domain
  such that for each pair of elements there is a pair
  such that & .

Remark:

• A principal ideal domain is a Bézout domain which satisfies the divisor chain condition.

Theorem 2.9

If is a discrete domain then the following are equivalent:

1. is a Bézout domain
2. Every finitely generated ideal in is principal
3. Every finitely generated ideal in R is principal, and is a GCD-domain

possible project?

Corollary 2.10

If is a discrete field then is a bounded principal ideal domain

Divisibility in discrete domains

3. Polynomial Rings

HEGL Illustrating Mathematics Seminar

Winter Semester 2024-2025

by Alina Stock

pseudonorm

• a function from nonzero elements of a discrete domain R to the nonnegative integers
• is a pseudonorm if for, a and b nonzero elements of R with a|b, then either or there exists such that .

Dedekind-Hasse map

• a function from nonzero elements of a discrete domain R to the nonnegative integers
• is a Dedekind-Hasse map if for any nonzero elements a and b of R either a|b, or there exists nonzero r in (a,b) such that

Euclidian map

• a function from nonzero elements of a discrete domain R to the nonnegative integers
• is a euclidean map if for any nonzero elements and b of R either a|b, or there exists nonzero r in R such that a|(b-r) and .

multiplicative

• is multiplicative if for all nonzero
• a multiplicative pseudonorm is called a multiplicative norm.

Polynomial Rings

Definition of content and primitive

• Let R be a GCD-domain, and let .

• The GCD of the coefficients of f is called the content of f, denoted by cont(f)

• If cont(f)=1, then f is said to be primitive.

Gauss's lemma

• Let R be a GCD-domain

• Let .

• Then cont(fg)=cont(f)cont(g)

Theorem Kronecker 1

If R is an infinite UFD with finitely many units, then so is R[X]. Thus R is factorial.

Unique interpolation (II, 5.5)

• Let be distinct elements of a field K
• Let be any n+1 elements of K
• Then there is a unique polynomial of degree at most n such that for all i.

Theorem Kronecker 2

If R is a factorial domain, then so is R[X].

Lemma (4.2)

• R a GCD-domain with fraction field K.
• If , then we can find and primitive such that f=cg.
• If f= for and primitive , then for some unit .

Corollary (4.4)

• R a GCD-domain with fraction field K, f and g polynomials in R[X]
• Then f divides g in R[X] if and only if f divides g in K[X] and cont(f) divides cont(g).