Principal Ideal Domain in Discrete mathematics

The PID can be described as an integral domain in which a single element is used to generate every proper ideal. To understand the PID, we have to first learn about the algebraic structure and Rings. After that, we will be able to understand the term integral domain, Principal ideal, and principal ideal domain.

Algebraic structure

An algebraic structure can be described as a non-empty set G which is equipped with one or more than one binary operation. In the binary operation, we take two inputs and perform an operation such as addition, subtraction, division, or multiplication on these inputs. After this operation, we will get a number as an output. Here (R, +, *) is known as the algebraic structure, which is equipped with two operations, i.e., addition (+) and multiplication (*).

Example:

1. (N, +), where N is used to indicate the set of natural numbers
2. (R, *), where R is used to indicate the set of real numbers, and * is used to indicate a multiplication operation.

Ring

A ring can be formed with the help of an algebraic structure, which is used to set the processing of two binary operations simultaneously. When we perform some operations like addition or multiplication on a nonempty set R, then it will be known as a ring if it contains the following conditions:

1. (R, +) will be known as the abelian group when it satisfies various groups such as G1, G2, G3, G4, and G5.
2. (R, *) will be known as Semi-group when it satisfies various groups such as G1, and G2.
3. Multiplication is distributive over addition:
   1. Right distributive: (y + z) * x = (y * x) + (z * x) ; ∀ x, y, z ∈ Rb)
   2. Left distributive: (y + z) * x = (y * x) + (z * x) ; ∀ x, y, z ∈ R

A detailed explanation of these is described as follows:

Group:

An algebraic structure (G, o) will be a group if o satisfies all the below conditions. Here G is used to indicate a non-empty set, and o is used to indicate the binary operation. These conditions are described as follows:

G1: Group G1 will be known as the closure if it contains conditions such as x ∈ G, y ∈ G => x o y ∈ G, here x, y ∈ G.

G2: Group G2 will be known as the associative if it contains conditions such as (x o y) o z = x o (y o z), where x, y, z ∈ G.

G3: Group G3 will be known as the identity element if it contains conditions like x o e = e o x = x, here x ∈ G. Here, e is used as an identity. For example: The identity will be 0 for the addition.

G4: Group G4 will be known as the Existence of Inverse if it contains conditions like x o x^-1 = x^-1 o x = e for x ∈ G.

Abelian Group

An algebraic structure (G, o) will be an Abelian group if o satisfies all the below properties of G1, G2, G3, G4, and one more additional G5. Here G is used to indicate a non-empty set, and o is used to indicate the binary operation. These conditions are described as follows:

G1: It is closure because it satisfies conditions like x ∈ G, y ∈ G => x o y ∈ G, where x, y ∈ G.

G2: It is associative because it satisfies conditions like (x o y) o z = x o (y o z), where x, y, z ∈ G.

G3: It is an identity element because it satisfies conditions like x o e = e o x = x, where x ∈ G. Here e is used as an identity. For example: The identity will be 0 for the addition.

G4: It is Existence of Inverse because it satisfies conditions like x o x^-1 = x^-1 o x = e for x ∈ G.

G5: It is Commutative because it satisfies conditions like x o y = y o x for x, y ∈ G.

Semi Group:

An algebraic structure (G, o) will be a semi-group if o satisfies all the below properties of G1 and G2. Here G is used to indicate a non-empty set, and o is used to indicate the binary operation. These conditions are described as follows:

G1: It is closure because it satisfies conditions like x ∈ G, y ∈ G => x o y ∈ G, where x, y ∈ G.

G2: It is associative because it satisfies conditions like (x o y) o z = x o (y o z), where x, y, z ∈ G.

There are two ways to write the ring structure, i.e., we can simply write R, or write (R, +, *).

Note: 0 is an additive unique quantity, which is also known as the zero elements of Ring R.

Commutative:

A ring R is commutative, which means that multiplication (*) is commutative.

Integral Domain

An integral domain can be described as a ring that contains (R, +, *). Here

(R, +, *) is commutative

It is used to specify that multiplication (*) is commutative.

(R,+,*) is a ring with a unit element

It is used to specify that there is a unit element, say 1∈ R in the below form:

R is a ring without zero divisors

For this, it will be in the following form:

Principal Ideal

Suppose (R, +, *) is a kind of commutative ring that contains identity 1.

Suppose that x ∈ R, then the set { ra : r ∈ R is an ideal } will be known as the principal ideal domain, which is generated by x.

Principal Ideal Domain (PID)

A ring R or (R, +, *) will be known as a principal ideal domain if it contains the following things:

• R must be an integral domain.
• In a ring R, every ideal is principal.

The ring will be termed into a primary ideal ring if each and every one-sided ideal of a ring is ideal. The PID can be described as a principal ring that contains no zero divisors.

Note: integrally closed domains ⊂ integral domains ⊂ commutative rings ⊂ rings.

Examples of PID (Principal Ideal Domain)

Example 1:

In this example, we have to show that every field is a Principal ideal domain.

Solution: For this, we will assume F as a field. Therefore, F is also an integral domain. This field F will also contain some unity element which is described as follows:

So, we can say that F is an integral domain with unity.

There are only 2 ideals for every field. That's why field F contains two ideals: F & {0}, which is described as follows:

So there are 2 ideals in F, and we can express these ideals with the help of a form, which is described as follows:

So, we can say that every field F is a Principal ideal domain (PID).

Note: The converse of this statement will not be true.

Example 2:

In this example, we have to prove that the ring of integers Z is a Principal ideal domain.

Solution: We know that the set of integers Z is a type of integral domain.

Suppose J is ideal in Z. Here, we have to show J is a principal ideal.

Case 1: If J = {0}, then it will be PI (principal ideal), and hence it will also be the result.

Case 2: Suppose 0 ≠ x ∈ J. If J ≠ {0}, then -x = {-1} x ∈ J for some positive x.

Hence, J is used to have a minimum of one positive integer. Suppose 'a' is the smallest positive integer in J.

Here we will claim that J = { ra : r ∈ Z}.

For x ∈ J, with the help of division algorithm, we will get the following:

But J is an ideal and a ∈ J, q ∈ Z.

But 'a' is used to show the smallest positive integer in J, which satisfies 0 ≤ r ≤ a. Hence, we will contain r = 0.

Hence, Z is a Principal Ideal Domain (PID).