Euclidean Domain and Congurence Problems

I took Abstract Algebra in UW-Madison(MATH541) with professor Chenxi Wu. 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.

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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.

Principle Ideal Domain (PID)

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

Definition 4. [Principle Ideal, Generating Set]

1. Let be a communicative ring with identity. An ideal is called a principle 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. [Principle Ideal Domain(PID), Greatest Common Divisor]

1. An integral domain with every ideal is principal is called principle ideal domain(PID).
2. From Theorem 3, we have for any , . is called the greatest common divisor of and , denoted as .

Definition 6. [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;

Unique Factorization

Definition 7. [Unit, Prime]

Let be a communicative ring with identity.

1. is a unit if it has multiplicative inverse.
2. is a prime if is a prime ideal. That is, is an integral domain.