Principal Ideal Domains

1. Formula
   Observation

   I

2. Audio
   Observation

   Let I be an ideal in R.

I

An ideal in a ring R.

Ideal of R

1. Formula
   Observation

   R

2. Audio
   Observation

   Let I be an ideal in R.

R

A ring; later specified as a commutative ring, integral domain, or principal ideal domain depending on the context.

Ring

1. Formula
   Observation

   $c \in R$

2. Audio
   Observation

   for some c in R.

c

An element of R that generates the principal ideal I.

Element of R

1. Formula
   Observation

   a, $b \in R \setminus$ \{0\}

2. Audio
   Observation

   Suppose that R is a commutative ring and a and b are in R without zero.

a, b

Non-zero elements of the ring R.

$R \setminus$ \{0\}

1. Formula
   Observation

   $d \in R$

2. Audio
   Observation

   We say that d in R is a greatest common divisor...

d

A greatest common divisor of a and b.

Element of R

1. Formula
   Observation

   $r \in R$

2. Audio
   Observation

   if r in R divides both a and b, then r divides d.

r

An arbitrary element of R used in the universal condition for the greatest common divisor.

Element of R

1. Formula
   Observation

   $l \in R$

2. Audio
   Observation

   We say that l in R is a least common multiple...

l

A least common multiple of a and b.

Element of R

1. Formula
   Observation

   mR

2. Audio
   Observation

   where m is an LCM of a and b.

m

A least common multiple of a and b in a PID.

Element of R

1. Formula
   Observation

   $I = (c) \text{ for } c \in R$

2. Audio
   Observation

   We say that I is a principal ideal if I is equal to the ideal generated by c for some c in R.

A principal ideal is an ideal I in a ring R that can be generated by a single element c from R.

1. I is an ideal in R

2. c is an element of R

1. Formula
   Observation

   $\text{Principal ideal domain (PID) integral domain for which every ideal principal}$

2. Audio
   Observation

   A principal ideal domain or PID is an integral domain for which every ideal is principal.

A principal ideal domain is an integral domain where every ideal is a principal ideal.

1. R is an integral domain

2. Every ideal in R is principal

1. Definition of Principal Ideal

1. Formula
   Observation

   $d \text{ divides both } a \text{ and } b$, $\text{ and if } r \in R \text{ divides both } a \text{ and } b$, $r \text{ divides } d$

2. Audio
   Observation

   We say that d in R is a greatest common divisor or GCD of a and b if d divides both a and b, and if r in R divides both a and b, then r divides d.

In a commutative ring, a greatest common divisor of two non-zero elements is an element that divides both of them, and is divisible by any other element that divides both.

1. R is a commutative ring

2. a, $b \in R \setminus$ \{0\}

3. $d \in R$

1. Formula
   Observation

   $a \text{ and } b \text{ both divide } l$, $\text{ and for all } r \in R$, $\text{ if } a \text{ and } b \text{ both divide } r$, $l \text{ divides } r$

2. Audio
   Observation

   We say that l in R is a least common multiple or LCM of a and b if a and b both divide l, and for all r in R, if a and b both divide r, then l divides r.

In a commutative ring, a least common multiple of two non-zero elements is an element that is divided by both of them, and divides any other element that is divided by both.

1. R is a commutative ring

2. a, $b \in R \setminus$ \{0\}

3. $l \in R$

1. Formula
   Observation

   $\text{In commutative ring, greatest common divisors may not exist and, if do exist, may not unique}$

2. Audio
   Observation

   In an arbitrary ring, which is commutative, greatest common divisors may not exist, and if they do exist, they may not be unique.

In a general commutative ring, greatest common divisors may not exist, and if they do exist, they may not be unique.

1. R is a commutative ring

For arbitrary commutative rings.

1. Formula
   Observation

   $\text{If } d \text{ gcd of } a \text{ and } b$, $\text{ every associate of } d \text{ gcd of } a \text{ and } b$

2. Audio
   Observation

   If d is a GCD of a and b, then every associate of d is also a GCD of a and b.

If d is a greatest common divisor of a and b, then every associate of d is also a greatest common divisor of a and b.

1. R is a commutative ring

2. a, $b \in R \setminus$ \{0\}

3. d is a GCD of a and b

For all associates of d.

1. Formula
   Observation

   $\text{If } R \text{ integral domain all gcds of } a \text{ and } b \text{ associates}$

2. Audio
   Observation

   If R is an integral domain, then all GCDs of a and b are associates.

If R is an integral domain, then all greatest common divisors of a and b are associates.

1. R is an integral domain

2. a, $b \in R \setminus$ \{0\}

For all GCDs of a and b.

1. Formula
   Observation

   $\text{In general lcms not unique}$

2. Audio
   Observation

   Just as for GCDs, in general LCMs are not unique.

In general, least common multiples are not unique.

1. R is a commutative ring

For arbitrary commutative rings.

1. Formula
   Observation

   aR + bR = dR $\text{ where } d \text{ gcd of } a \text{ and } b$

2. Audio
   Observation

   Then aR plus bR is equal to dR where d is a GCD of a and b.

Let R be a principal ideal domain and let a, b be non-zero elements of R. Then the sum of the ideals generated by a and b is equal to the ideal generated by a greatest common divisor of a and b.

1. R is a PID

2. a, $b \in R \setminus$ \{0\}

For all non-zero a, b in R.

1. Formula
   Observation

   aR $\cap$ bR = mR $\text{ where } m \text{ lcm of } a \text{ and } b$

2. Audio
   Observation

   And aR intersect bR is equal to mR where m is an LCM of a and b.

Let R be a principal ideal domain and let a, b be non-zero elements of R. Then the intersection of the ideals generated by a and b is equal to the ideal generated by a least common multiple of a and b.

1. R is a PID

2. a, $b \in R \setminus$ \{0\}

For all non-zero a, b in R.

1. Formula
   Observation

   This formula needs review. Please report it using the page feedback control.{ UFD

2. Audio
   Observation

   If R is a PID, then R is a UFD.

If R is a principal ideal domain, then R is a unique factorization domain.

1. R is a PID

For all PIDs.

1. Diagram
   Observation

   Title card with text 'Principal Ideal Domains', 'Algebra', 'Ep. 18'.

1. Text boxes

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1. Text content

Introduction to the topic of Principal Ideal Domains.

1. Diagram
   Observation

   Slide titled 'Principal Ideal' with bullet points defining it.

1. Title box

2. Bullet points

1. Bullet points appear sequentially

1. Slide title

Visual presentation of the definition of a principal ideal.

1. Diagram
   Observation

   Slide titled 'Principal Ideal Domain' with a bullet point defining it.

1. Title box

2. Bullet point

1. Bullet point appears

1. Slide title

Visual presentation of the definition of a principal ideal domain.

1. Diagram
   Observation

   Slide titled 'Greatest Common Divisor' with bullet points defining it and noting existence/uniqueness issues.

1. Title box

2. Bullet points

1. Bullet points appear sequentially

1. Slide title

Visual presentation of the definition and properties of the greatest common divisor.

1. Diagram
   Observation

   Slide titled 'How Gcds & Associates Relate' with bullet points explaining the relationship.

1. Title box

2. Bullet points

1. Bullet points appear sequentially

1. Slide title

Visual presentation of how greatest common divisors relate to associates.

1. Diagram
   Observation

   Slide titled 'Lowest Common Multiple' with bullet points defining it and noting uniqueness issues.

1. Title box

2. Bullet points

1. Bullet points appear sequentially

1. Slide title

Visual presentation of the definition and properties of the lowest common multiple.

1. Diagram
   Observation

   Slide titled 'How Pids Relates To Gcds & Lcms' with bullet points showing ideal operations.

1. Title box

2. Bullet points

1. Bullet points appear sequentially

1. Slide title

Visual presentation of how principal ideal domains relate to greatest common divisors and lowest common multiples via ideal operations.

1. Diagram
   Observation

   Slide titled 'Remark For Pids' with a bullet point stating PID implies UFD.

1. Title box

2. Bullet point

1. Bullet point appears

1. Slide title

Visual presentation of a key property of principal ideal domains.

1. Diagram
   Observation

   Outro card with text 'Thank You For Watching'.

1. Text box

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1. Text content

Conclusion of the video.

1. Formula
   Observation

   $\text{greatest common divisors may not exist and, if do exist, may not unique}$

2. Audio
   Observation

   greatest common divisors may not exist, and if they do exist, they may not be unique.

Assuming that greatest common divisors always exist and are unique in any commutative ring.

In a general commutative ring, GCDs may not exist, and even when they do, they are not necessarily unique.

1. Formula
   Observation

   $\text{In general lcms not unique}$

2. Audio
   Observation

   in general LCMs are not unique.

Assuming that least common multiples are always unique in any commutative ring.

Similar to GCDs, least common multiples are not generally unique in an arbitrary commutative ring.

1. Audio
   Observation

   A principal ideal domain or PID is an integral domain for which every ideal is principal.

The concept of a principal ideal is required to define a principal ideal domain.

1. Audio
   Observation

   Let R be a PID... Then aR plus bR is equal to dR where d is a GCD of a and b.

The property of being a principal ideal domain is applied to establish the relationship between the sum of ideals and the GCD.

1. Audio
   Observation

   Let R be a PID... And aR intersect bR is equal to mR where m is an LCM of a and b.

The property of being a principal ideal domain is applied to establish the relationship between the intersection of ideals and the LCM.

1. Audio
   Observation

   Next, let's discuss how PIDs relate to GCDs and LCMs.

GCDs and LCMs are dual concepts in ring theory, often discussed together in the context of PIDs.

1. Audio
   Observation

   First, let's define a principal ideal.

1. Definition of Principal Ideal

1. Audio
   Observation

   Now, let's define the principal ideal domain.

1. Definition of Principal Ideal Domain

1. Audio
   Observation

   Next, let's define the greatest common divisor.

1. Definition of Greatest Common Divisor

1. Audio
   Observation

   In an arbitrary ring... greatest common divisors may not exist, and if they do exist, they may not be unique.

1. Existence and Uniqueness of GCDs in Commutative Rings

1. Audio
   Observation

   Then, let's talk about how GCDs and associates relate.

1. Associates of a GCD are GCDs
2. All GCDs in an Integral Domain are Associates

1. Audio
   Observation

   Following on, let's define the lowest common multiple.

1. Definition of Least Common Multiple

1. Audio
   Observation

   Next, let's discuss how PIDs relate to GCDs and LCMs.

1. Sum of Principal Ideals and GCD in a PID
2. Intersection of Principal Ideals and LCM in a PID

1. Audio
   Observation

   Now, let's discuss a quick remark for PIDs.

1. Every PID is a UFD