FoldUnfold Table of Contents The Inclusion Chart for Special Types of Integral Domains The Inclusion Chart for Special Types of Integral Domains Below is an inclusion chart showing the various inclusions of the algebraic structures of: commutative rings, integral domains, unique factorization domains, principal ideal domains, Euclidean domains, and … [Read more...]

## Unique Factorization Domains (UFDs)

FoldUnfold Table of Contents Unique Factorization Domains (UFDs) Unique Factorization Domains (UFDs) Definition: Let $(R, +, \cdot)$ be an integral domain. Then $R$ is a Unique Factorization Domain if the following properties are satisfied: 1) Every element $a \in R$ that is nonzero and that is not a unit can be expressed as a product of irreducible elements in $R$. 2) … [Read more...]

## Irreducible Elements in a Commutative Ring

FoldUnfold Table of Contents Irreducible Elements in a Commutative Ring Irreducible Elements in a Commutative Ring Recall from The Greatest Common Divisor of Elements in a Commutative Ring page that if $(R, +, \cdot)$ is a commutative ring and $a_1, a_2, ..., a_n \in R$ then a greatest common divisor of these elements is an element $d \in R$ which satisfies the following … [Read more...]

## The Greatest Common Divisor of Elements in a Commutative Ring

FoldUnfold Table of Contents The Greatest Common Divisor of Elements in a Commutative Ring The Greatest Common Divisor of Elements in a Commutative Ring Recall from the Divisors and Associates of Commutative Rings page that if $(R, +, \cdot)$ is a commutative ring then for $a, b \in R$ we said that $b$ is a divisor of $a$ written $b | a$ if there exists an element $q \in … [Read more...]

## Associates of Elements in Commutative Rings

FoldUnfold Table of Contents Associates of Elements in Commutative Rings Associates of Elements in Commutative Rings Definition: Let $(R, +, \cdot)$ be a commutative ring and let $a, b \in R$. Then $a$ is said to be an Associate of $b$ denoted $a \sim b$ if there exists a unit $u \in R$ such that $a = bu$. Theorem 1: If $(R, +, \cdot)$ is an integral domain and $a … [Read more...]

## Divisors of Elements in Commutative Rings

FoldUnfold Table of Contents Divisors of Elements in Commutative Rings Divisors of Elements in Commutative Rings So far we have discussed the term "divisor" with regards to integers and polynomials over a field $F$. We will now extend the notion of a divisor to a general commutative ring. Definition: Let $(R, +, \cdot)$ be a commutative ring and let $a, b \in R$. Then … [Read more...]

## The Set of Units of a Ring forms a Group under *

FoldUnfold Table of Contents The Set of Units of a Ring forms a Group under * The Set of Units of a Ring forms a Group under * Recall from the Units (Multiplicatively Invertible Elements) in Rings page that if $(R, +, *)$ is a ring with multiplicative identity $1$ then an element $a \in R$ is said to be a unit (or a multiplicatively invertible element) in $R$ if there … [Read more...]

## The Set of Units in Mnn

FoldUnfold Table of Contents The Set of Units in Mnn The Set of Units in Mnn Recall from the Units (Multiplicatively Invertible Elements) in Rings page that if $(R, +, *)$ is a ring then an element $a \in R$ is said to be unit or a multiplicatively invertible element if there exists an element $b \in R$ such that $a * b = 1$ and $b * a = 1$ where we denote $b = … [Read more...]

## Units (Multiplicatively Invertible Elements) in Rings

FoldUnfold Table of Contents Units (Multiplicatively Invertible Elements) in Rings Units (Multiplicatively Invertible Elements) in Rings Definition: Let $(R, +, *)$ be a ring with identity $1$. An element $a \in R$ is said to be a Unit or Multiplicatively Invertible Element if there exists an element $b \in R$ such that $a * b = 1$ and $b * a = 1$. If such a $b$ … [Read more...]

## Every Euclidean Domain is a Principal Ideal Domain

FoldUnfold Table of Contents Every Euclidean Domain is a Principal Ideal Domain Every Euclidean Domain is a Principal Ideal Domain Recall from the Euclidean Domains (EDs) page that if $(R, +, *)$ is an integral domain then $R$ is said to be a Euclidean domain if there exists a function $\delta : R \setminus \{ 0 \} \to \mathbb{N} \cup \{ 0 \}$ that satisfies the following … [Read more...]