Euclidean / Principal Ideal / Unique Factorization Domain

I took Abstract Algebra I/II in UW-Madison(MATH541) with professor Chenxi Wu and Tonghai Yang. This article is inspired by prof. Wu's lecture note, and many formats, content, and structure of this blog are influenced by and similar to it.

RECAP

Please read this introduction with a complete understanding of ring.

Euclidean Domains

Definition 1. [Euclidean domain, Euclidean function]

An integral domain is Euclidean if there is a map such that for any and , such that (1) , and (2) or . is called Euclidean domain and is called Euclidean function.

Example 2. is an Euclidean domain as is a Euclidean function.

Theorem 3. Let be an Euclidean domain, then any ideal of can be written as for some .

Definition 4. [Principal Ideal, Generating Set]

1. Let be a communicative ring with identity. An ideal is called a principal ideal, written as , with called the generator;
2. More generally, let be the index set, , then the ideal consists all of the elements of the form , with , and all but finitely . Then, is called the ideal generated by , written as , and is called its generating set;

Definition 5. [Principal Ideal Domain(PID), Greatest Common Divisor]

1. An integral domain with every ideal is principal is called principal ideal domain(PID).

Definition 6. [Multiple/Divide/Divisor, Greatest Common Divisor]

1. is said to be a multiple of iff there is another element with . In this case, we say divide , or is a divisor of .
2. From Theorem 3, we have for any , . is called the greatest common divisor of and , denoted as .

Definition 7. [Coprime] Let be a communicative ring with identity 1. For two ideals , of , we called and are coprime if . For , if , we called and coprime;

Proposition 1. If are non-zero elements in the communicative ring such that the ideal generated by and (that is, ) is a principal ideal , then is a greatest common divisor of and .

Definition 8. [Unit, Prime Ideal]

Let be a communicative ring with identity.

1. is a unit if it has multiplicative inverse.
2. An ideal is a prime ideal if and only if , at least one of and is also in .

Proposition 9. If , we have for some unit . This gives if and are both g.c.d. of and , we have for some unit .

Theorem 10. Let be a Euclidean Domain and let and be nonzero elements of . Let be the last non-zero remainder in the Euclidean Algorithm for and , then we have

1. is a g.c.d. of and ;
2. can be written as an R-linear combination of and . There are such that

Principal Ideal Domains (PID)

Definition 11. [Principal Ideal Domain (P.I.D.)]

1. An integral domain is Principal Ideal Domain if and only if every ideal is principal.

Some examples:

1. is a PID.
2. is not PID. A counterexample is , which is a nonprincipal ideal.

Proposition 12. Every nonzero prime ideal in a Principal Ideal Domain is a maximal ideal.

Corollary 13. If is any communicative ring such that the polynomiall ring is a PID or Euclidean Domain, then is a field.

Definition 14. [Dedekind-Hasse Norm]

1. We define a norm to be a Dedekind-Hasse norm if is a positive norm and for every non-zero either is an element of the ideal or there exists with .

Proposition 15. Integral domain is a PID if and only if has a Dedekind-Hass Norm.

Unique Factorization Domain (UFD)

In terms of the integers , we can factorize any integer into primes . This property can be extended to a much larger class of rings, namely Unique Factorization Domain. To introduce the formal definition of UFD, let's introduce some concepts shortly.

Definition 16. [Irreducible/Reducible, Prime, Associate]

Let be an integral domain.

1. Let be a non-unit element. We say is reducible if and only if there are two non-unit elements such that . Otherwise, we say is irreducible if and only if such that , at least one of is a unit.
2. A non-zero element is called prime if and only if is a prime ideal. That is, for any , either or .
3. Two elements are said to be associate if and only if there exists a unit such that .

Proposition 17. In an integral domain, every prime element is irreducible.

Proposition 18. In a PID, a non-zero element is prime if and only if it is irreducible.

Definition 19. Unique Factorization Domain (UFD)

An integral domain is an Unique Factorization Domain (UFD) if every nonzero and nonunit element has the following two properties:

1. can be written as a finite product of irreducibles ;
2. For any other factorization of into irreducibles , we have , and we can renumber the factors so that is associate to for .

Proposition 20. In a UFD, a nonzero element is prime if and only if it is irreducible.

Proposition 21. Let R be an UFD, , suppose

are prime factorizations for and , where and are units, and the primes are distinct and . Then, we have

is a GCD of and .

Theorem 22. Every PID is a UFD. In particular, every Euclidean Domain is a UFD.

Corollary 23. (Fundamental Theorem of Arithmetic) The integer ring is a UFD.

Corollary 24. Let be a PID, then there exists a multiplicative Dedekind-Hasse norm on .