Integral Domain in Discrete Mathematics

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We know that rings and groups are fundamental algebraic structures that have a wide range of applications in discrete mathematics. Within rings, we have a special type known as the integral domain and they are quite useful.

If you are preparing for competitive exams or trying to learn advanced topics in algebra, then it is important that you understand what an integral domain is and how it differs from a general ring. Read this chapter to learn the basics of integral domain and its important properties.

What is a Ring in Group Theory?

For a basic recap, a ring is nothing but an algebraic structure consisting of a set R with two binary operations, these are addition (+) and multiplication (·). To form a ring, a set and its operations must satisfy certain conditions.

Addition forms an Abelian group, which indicates that adding any two elements of R results in another element from R. Associativity ensures that grouping of elements does not affect the result. and there must be an additive identity, which is typically 0. So, a + 0 = a. Every element must have an inverse under addition, and addition must be commutative.

The multiplication forms a semi-group, which satisfying closure, associativity, and a distribution property. For example, a ⋅ (b + c) = a ⋅ b + a ⋅ c. The multiplication operation must distribute over addition, resulting in a ring.

Now we must also recap the idea of ring with unity. This has a multiplicative identity element. It is denoted as 1, such that a ⋅ 1 = a = 1 ⋅ a for all a ∈ R.

What is an Integral Domain?

An integral domain is a type of ring with additional restrictions. A ring D is considered an integral domain if it satisfies the following:

Commutativity of Multiplication

In an integral domain, the multiplication operation must be commutative. This means for any two elements a and b in D,

$$\mathrm{a \:\cdot\:b \:=\: b \:\cdot\: a}$$

Not all rings require commutative multiplication, but in an integral domain, it is required.

No Zero Divisors

A ring has zero divisors when the product of two non-zero elements equals zero. For example, in some rings, a ⋅ b = 0 can occur even when both a and b are non-zero. So in an integral domain, this cannot happen.

If the product of two elements in an integral domain is zero, at least one of the elements must be zero. In other words, the ring D must satisfy −

$$\mathrm{a \:\cdot\: b \:=\: 0 \:\Rightarrow\: a \:=\: 0 \text{ or } b \:=\: 0}$$

This property is what distinguishes an integral domain from other types of rings.

Multiplicative Identity (Unity)

Like rings with unity, an integral domain must have a multiplicative identity as well. This element, often denoted as 1. This gives a ⋅ 1 = a for all a ∈ D.

From these properties we can say an integral domain is a commutative ring with unity that has no zero divisors.

Examples of Integral Domains

Let us understand more about Integral Domains with the help of following examples –

Example 1: Integers (Z)

The set of integers (Z) is the classic example of an integral domain.

• Commutativity − The multiplication of any two integers is commutative. For example, 3 ⋅ 5 = 5 ⋅
• No Zero Divisors − In the set of integers, the product of two non-zero elements is never zero. There is no way to multiply two non-zero integers and get zero.
• Unity − The integer 1 serves as the multiplicative identity, as a ⋅ 1 = a for any integer a.

Thus, Z satisfies all the properties of an integral domain.

Example 2: Polynomials with Real Coefficients (R[x])

The set of polynomials with real coefficients, denoted by R[x], is another example of an integral domain.

• Commutativity − Multiplying polynomials is commutative. For example, (x + 1)(x + 2) = (x + 2)(x + 1).
• No Zero Divisors − The product of two non-zero polynomials is never the zero polynomial. If the product is zero, then at least one of the polynomials must be zero.
• Unity − The polynomial 1 is the multiplicative identity, as multiplying any polynomial by 1 leaves it unchanged.

Therefore, R[x] is an integral domain.

Example 3: Integers Modulo a Prime Number (Z_p)

The set of integers modulo a prime number p, denoted as Z_p, is an integral domain. Because:

• Commutativity − Addition and multiplication modulo a prime number are commutative operations.
• No Zero Divisors − In Z_p, if p is prime, then the product of two non-zero elements modulo p is never zero. So that is one of the reasons why primes are so important in number theory. For instance, in Z₅, the product of any two non-zero elements modulo 5 is always non-zero.
• Unity − The number 1 serves as the multiplicative identity in Z_p.

Thus, Z_p is an integral domain.

Rings That Are Not Integral Domains

Let us see some examples of rings which are not falling under non-integral domains.

Example 1: Z₆ (Integers Modulo 6)

The set of integers modulo 6, denoted Z₆, is not an integral domain.

• Zero Divisors − In Z₆, we have zero divisors. For example, 2 ⋅ 3 = 0 mod 6, even though neither 2 nor 3 is zero.
• Commutativity and Unity − While multiplication is commutative and 1 is the multiplicative identity, the existence of zero divisors means that Z₆ is not an integral domain.

Example 2: Matrices

The set of 2 x 2 matrices with real entries, denoted M₂(R), is a ring but not an integral domain.

• Zero Divisors − In matrix multiplication. It is possible for two non-zero matrices to multiply and result in the zero matrix. This violates the requirement that an integral domain should have no zero divisors.
• Non-Commutativity − Matrix multiplication is not commutative. In general, for two matrices A and B, A ⋅ B ≠ B ⋅ A.

Thus, M₂(R) is not an integral domain.

Conclusion

An integral domain is a specific type of ring that satisfies additional properties beyond those of a basic ring. These properties include the absence of zero divisors, commutative multiplication, and the existence of a multiplicative identity.

In this chapter, we presented examples of integers (Z) and polynomials with real coefficients (R[x]) to understand what makes an integral domain distinct. We also saw examples of rings that do not qualify as integral domains, such as Z6 and matrix rings, due to the presence of zero divisors or non-commutative multiplication.