Rings and Subrings in Discrete Mathematics

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In Discrete Mathematics, groups, rings and fields are important concepts related to Algebraic Structures that extends the concept of groups. But what exactly is a ring, and how does it differ from groups or other algebraic structures like semigroups?

In this chapter, we will cover the concept of rings and subrings in detail. We will also discuss their properties and present several examples to for a better understanding.

What are Rings in Group Theory?

A ring is nothing but an algebraic structure made up with set of elements combined with two binary operations. They are usually referred to as addition (+) and multiplication (·). Unlike groups, where we have only one binary operation; rings are characterized by having two operations that must satisfy certain properties.

Ring Properties

For a set R to be considered a ring, it must satisfy the following properties −

Addition forms an Abelian group

The set R, under the operation of addition (+), must be forming Abelian group. This means:

• Closure − Adding any two elements of the set produces another element in the set. For example, if a, b ∈ R, then a + b ∈ R.
• Associativity − Addition must be associative, meaning (a + b) + c = a + (b + c) for all a, b, c ∈ R.
• Identity Element − There must exist an identity element for addition, often denoted by 0, such that a + 0 = a for all a ∈ R.
• Inverse Element − Every element must have an additive inverse, such that a + (−a) = 0.
• Commutativity − The addition operation is commutative, meaning a + b = b + a for all elements a, b ∈ R.

Multiplication forms a Semigroup

The set R, is under multiplication (·), they must satisfy the properties of a semigroup. That means:

• Closure − If a, b ∈ R, then a ⋅ b ∈ R.
• Associativity − Multiplication must be associative, meaning (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c).

Distributive Property

The multiplication operation must be distributive over addition. This means:

• Left distributivity − a ⋅ (b + c) = a ⋅ b + a ⋅ c for all a, b, c ∈
• Right distributivity − (a + b) ⋅ c = a ⋅ c + b ⋅ c for all a, b, c ∈

If a set satisfies these three properties, it is considered a ring.

Example of Integers Under Addition and Multiplication

Let us now look at some examples to understand the concept better. The set of integers (Z) with the operations of addition (+) and multiplication (·).

• Addition − The set of integers forms an Abelian group under addition. It satisfies the closure, associativity. It has an identity element (0), and every element has an additive inverse (Like the inverse of 5 is -5). The addition is also commutative, as 5 + 3 = 3 + 5.
• Multiplication − The set of integers generate a semigroup under multiplication. And it satisfies closure and associativity. But, multiplication in a general ring does not need to be commutative, though in this case, it is.
• Distributive property − Multiplication is distributive over addition. For example, 2 ⋅ (3 + 4) = 2 ⋅ 3 + 2 ⋅ 4 = 6 + 8 = 14.

Here, the set of integers with addition and multiplication forms a ring.

Types of Rings

After getting the basic idea on rings. Let us talk about the different types of them based on additional properties they may satisfy. Let us see some of the common types of rings.

Commutative Ring

A commutative ring is a ring where multiplication is also commutative. It means, a ⋅ b = b ⋅ a for all elements in the ring. The set of integers (Z) under addition and multiplication is an example of a commutative ring because multiplication in the integers is commutative (e.g., 3 ⋅ 4 = 4 ⋅ 3).

Ring with Unity

The ring with unity (also called a unital ring) is a ring that has a multiplicative identity element. It is usually denoted by 1. This element must satisfy the property that for any element a ∈ R, a ⋅ 1 = a = 1 ⋅ a.

In simple words, multiplying any element by 1 gives the element unchanged. For example, the set of integers (Z) is a ring with unity because 1 acts as the multiplicative identity.

Finite and Infinite Rings

A ring is called finite if it contains a finite number of elements. Otherwise, it is an infinite ring. So the set of integers (Z) is an example of an infinite ring. As there are infinitely many integers. But, a set like Z₆ (integers modulo 6) would form a finite ring since it only contains the elements {0, 1, 2, 3, 4, 5} only.

Ring with Zero Divisors

A ring with zero divisors is one where there exist non-zero elements a and b such that a ⋅ b = 0. In other words, multiplying two non-zero elements results in zero. The example of a ring with zero divisors is the set Z₆ (integers modulo 6). In this ring, 2 ⋅ 3 = 0, even though both 2 and 3 are non-zero elements.

Ring without Zero Divisors

A ring without zero divisors is called an integral domain. In such a ring, if a ⋅ b = 0, then either a = 0 or b = 0. The set of integers (Z) is an integral domain because if the product of two integers is zero, then at least one of the integers must be zero. We will learn this integral domain in detail later.

What is a Subring?

A subring is nothing but a subset of a ring that is itself a ring under the same operations as the original ring. To be a subring, the subset must satisfy the following conditions:

• It must be closed under both addition and multiplication.
• It must contain the additive identity (usually 0).
• It must include the additive inverses of all its elements.

Example of a Subring

Consider the ring of integers (Z) and its subset 2Z = {…, −4, −2, 0, 2, 4, …}, which consists of all even integers.

• Closed under addition − The sum of any two even integers is also an even integer. For example, 2 + 4 = 6, which is in the set.
• Closed under multiplication − The product of any two even integers is also an even integer. For example, 2 ⋅ 4 = 8, which is in the set.
• Contains additive identity − The element 0 is in the set 2Z.
• Contains additive inverses − For every even integer, its negative is also in the set. For example, the inverse of 4 is -4, and both are in 2Z.

Thus, 2Z is a subring of Z.

Conclusion

To conclude, a ring is an algebraic structure with two binary operations: addition and multiplication. A ring satisfies specific properties like being an Abelian group under addition and a semigroup under multiplication. Rings also respect the distributive property.

In this chapter, we covered various types of rings, including commutative rings, rings with unity, finite and infinite rings, and rings with or without zero divisors. Finally, we explained the concept of a subring, which is a subset of a ring that retains the ring's properties.