Properties of Rings in Discrete Mathematics

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Groups and Rings are quite similar but they have several different properties. A ring is an algebraic structure that defines operations within a set. However, unlike a group, a ring has two binary operations, typically addition and multiplication. These operations must satisfy several important properties for the structure to be classified as a ring.

In this chapter, we will explain the various properties of rings, using examples for a better understanding.

Rings and Their Properties

For a basic recap, a ring is nothing but an algebraic structure defined by a set R together with two binary operations, called addition (+) and multiplication (·). The set and these operations must follow a set of specific rules or properties to qualify as a ring.

Formally, a ring R must satisfy the following properties −

• Addition forms an Abelian group − The set R, under the operation of addition, must form a group where the addition is commutative. In other words, addition within the ring must be associative. It must have an identity element, and must include inverse elements. The order of addition does not matter.
• Multiplication forms a semigroup − The set R, under multiplication, must at least satisfy closure and associativity properties. However, multiplication need not to have an identity or inverses, and it need not to be commutative.
• Distributive property − Multiplication in a ring must distribute over addition. For any three elements a, b, and c in the ring, the following must hold −
  • a ⋅ (b + c) = a ⋅ b + a ⋅ c (left distributive)
  • (a + b) ⋅ c = a ⋅ c + b ⋅ c (right distributive)

Properties of Rings

Let us see the basic properties of rings in more detail −

Closure under Addition and Multiplication

From the basic definitions of rings, we know that for any two elements a and b in the ring R, both a + b and a ⋅ b must also belong to R. This is the closure property. It ensures that performing operations within the set does not give in elements outside the set.

For example, consider the set of integers Z with the standard operations of addition and multiplication. If we add or multiply any two integers, the result is always another integer, so the set of integers is closed under both addition and multiplication.

Associativity of Addition and Multiplication

The associative property indicates that, for any elements a, b, and c in the ring, the grouping of the elements during the operation does not affect the result. So this applies to both addition and multiplication.

• For addition: (a + b) + c = a + (b + c)
• For multiplication: (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c)

Again, the integers under addition and multiplication satisfy this property. Whether we add or multiply numbers in any grouping, the results will be same.

Additive Identity

The additive identity element is usually denoted as 0. This is the element that, when added to any element a in the ring, leaves a unchanged. In other words, for any a ∈ R,

$$\mathrm{a \:+\: 0 \:=\: a \:=\: 0 \:+\: a}$$

In the set of integers, the number 0 is the additive identity because adding 0 to any number does not change the number.

Additive Inverse

For every element a in a ring R, there must be an element b (called the additive inverse) so that a + b = 0, where 0 is the additive identity. And the additive inverse of a is usually denoted as (−a).

For example, in the set of integers, the additive inverse of 5 is -5, because 5 + (−5) = 0.

Distributive Property of Multiplication over Addition

Next property comes for the distributive nature. One of the defining properties of a ring is that multiplication must distribute over addition. So for all a, b, and c in the ring R,

$$\mathrm{a \:\cdot\: (b \:+\: c) \:=\: a \:\cdot\: b \:+\: a \:\cdot\: c \:=\: (b \:+\: c) \:\cdot\: a \:=\: b \:\cdot\: a \:+\: c \:\cdot\: a}$$

This property shows that multiplication is compatible with addition, which is needed for the algebraic structure of the ring.

Additional Ring Types

Rings can be classified into different types based on additional properties they might satisfy. Let us discuss on them briefly.

Commutative Rings

A commutative ring is a ring where multiplication is commutative. So that that for all a and b in the ring, the order of multiplication doesnt matter −

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

The set of integers under standard addition and multiplication is an example of a commutative ring because the order of multiplying two integers does not affect the result.

Ring with Unity

A ring with unity (also called a unital ring) is a ring that contains a multiplicative identity. It is denoted as 1. This element satisfies the property that for any element a ∈ R,

$$\mathrm{a \:\cdot\: 1 \:=\: a \:=\: 1 \:\cdot\: a}$$

In the integers, the number 1 is the multiplicative identity because multiplying any number by 1 will not change the number.

Integral Domain

An integral domain is a commutative ring with unity that has no zero divisors. As we know a zero divisor is a non-zero element a in the ring such that there exists another non-zero element b where a ⋅ b = 0.

For example, the set of integers is an integral domain because there are no non-zero integers a and b such that a ⋅ b = 0.

Ring with Zero Divisors

Next type is a ring with zero divisors. This is a ring where it is possible for the product of two non-zero elements to be zero. An example of this is the ring Z₆ (integers modulo 6). In this ring, 2 ⋅ 3 = 0 even though both 2 and 3 are non-zero elements.

Conclusion

A ring is an algebraic structure consisting of a set with two operations: addition and multiplication. In this chapter, we presented several essential properties of rings, including closure, associativity, the existence of an additive identity, and the distributive property.

We also looked at different types of rings such as commutative rings, rings with unity, and rings with zero divisors.