Ring Theory: Definition, Examples, Problems & Solutions

The ring theory in Mathematics is an important topic in the area of abstract algebra where we study sets equipped with two operations addition (+) and multiplication (⋅). In this article, we will study rings in abstract algebra along with its definition, examples, properties and solved problems.

Definition of a Ring

Let R be a non-empty set. A pair (R, +, ⋅) is called a ring if the following conditions are satisfied.

1. (R, +) is a commutative group.
2. (R, ⋅) is a semigroup.
3. Distributive laws hold on R.

Let us now elaborate these properties below.

Properties of a Ring

In a ring (R, +, ⋅) we have that R is an abelian group under +, R is a semigroup under ⋅ and both the distributive laws hold on R. More specifically, (R, +, ⋅) satisfies the following properties.

(R, +) is an abelian group	(i) [Closure]: for all a, b ∈ R, we have a+b ∈ R. (ii) [Associativity]: a+(b+c) = (a+b)+c ∀ a, b, c ∈ R. (iii) [Identity]: there exists an element o ∈ R such that a+o = o+a = a ∀ a ∈ R. The element o is called the additive identity in R. (iv) [Inverse]: for every a ∈ R there exists an element b ∈ R such that a+b = b+a = o, the additive identity. The element b is called an additive inverse of a in R. (v) [Commutativity]: a+b = b+a ∀ a, b ∈ R.
(R, ⋅) is a semigroup	(vi) [Closure]: for all a, b ∈ R, we have a⋅b ∈ R. (vii) [Associativity]: a⋅(b⋅c) = (a⋅b)⋅c ∀ a, b, c ∈ R.
Distributive laws	(viii) For all a, b, c ∈ R, we have that a⋅(b+c) = a⋅b + a⋅c [left distributive law] (a+b)⋅c = a⋅c + b⋅c [right distributive law]

Types of Rings

There are many types of rings. For example,

1. Zero Ring or Null Ring: The set ({0}, +, ⋅) forms a ring with the operations 0+0=0 and 0⋅0=0. This is called a zero/trivial ring.
2. Ring With Unity: A ring (R, +, ⋅) is called a ring with unity if the multiplicative inverse exists in R. That is, there exists an element e ∈ R such that a⋅e = e⋅a = a ∀ a ∈ R.
3. Ring Without Unity: A ring (R, +, ⋅) is called a ring without unity if the multiplicative inverse does not exist in R.
4. Commutative Ring: A ring (R, +, ⋅) is called a commutative/abelian ring if (R, ⋅) is commutative. In other words, a⋅b = b⋅a ∀ a, b ∈ R.
5. Non-Commutative Ring: A ring (R, +, ⋅) is called a non-commutative ring if (R, ⋅) is not commutative.
6. Boolean Ring: A ring R is called a boolean ring if a²=a ∀ a ∈ R, that is, every element is idempotent.
7. Simple Ring: A ring R is called a simple ring if it has no non-trivial proper ideals.

More properties of a ring:

• In a commutative ring, the two (left and right) distributive laws are the same.
• The additive identity element in a ring R is also called the zero element in R, and it is usually denoted by 0.
• The multiplicative identity (if exists) is unique in a ring, and it is usually denoted by I.
• If R is a non-trivial ring with unity I, then I ≠ 0.

Examples of Rings

A few examples of rings in Mathematics are given below.

1. (ℝ, +, ⋅), (ℚ, +, ⋅), (ℤ, +, ⋅) are rings. They are commutative rings without unity 1.
2. (ℤ_n, +, ⋅) is a commutative ring with unity $\overline{1}$. This is called the ring of integers modulo n.
3. (2ℤ, +, ⋅) is a commutative ring, but the multiplicative identity doest not exists. More generally, (nℤ, +, ⋅) is an example of a commutative ring without unity where n is a positive integer.
4. (M₂(ℝ), +, ⋅) is a non-commutative ring with unity (identity matrix) where M₂(ℝ) is the set of all 2×2 matrices with real entries.
5. Ring of Gaussian integers: Let ℤ[i] = {a+bi : a, b ∈ ℤ} be the set of all complex numbers of the form a+ib where a and b are integers. (ℤ[i], +, ⋅) is a commutative ring of unity called the ring of Gaussian integers.
6. Ring of Gaussian numbers: Let ℚ[i] = {a+bi : a, b ∈ ℚ} be the set of all complex numbers of the form a+ib where a and b are rational numbers. Note that (ℚ[i], +, ⋅) forms a commutative ring of unity, called the ring of Gaussian numbers.
7. Polynomial ring: For a ring R, the set R[x] = {a₀+a₁x+…+a_nxⁿ : a_i ∈ R} of all polynomials over R form a ring unity. This is called the polynomial ring over R.

Solved Problems

Question 1: Give an example of a commutative ring with unity.

Answer:

Let ℝ^× = ℝ – {0}. Then the ring (ℝ^×, +, ⋅) is a commutative ring with unity 1.

Read Also: Units in a Ring: Definition, Examples, How to Find

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