Rings and Fields

 

Rings in Discrete Structures

A ring is an algebraic structure that extends the concept of a group by introducing a second operation. It consists of a set equipped with two operations (usually addition and multiplication) that satisfy specific properties.

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Definition of a Ring

A *ring (R, +, ) is a set R with two binary operations:

1. Addition (+): Forms an Abelian group.
2. Multiplication (*): Is associative and distributes over addition.

Properties of a Ring

A set $R$ with operations $+$ and $*$ is a ring if it satisfies the following:

1. (R, +) is an Abelian Group

• Closure: $a + b \in R$ for all $a, b \in R$.
• Associativity: $(a + b) + c = a + (b + c)$.
• Identity Element (Additive Identity): There exists an element 0 such that $a + 0 = a$.
• Inverse Element (Additive Inverse): For every $a \in R$, there exists $-a$ such that $a + (-a) = 0$.
• Commutativity: $a + b = b + a$ for all $a, b \in R$.

*2. (R, ) is a Semigroup (Associative)

• Closure: $a * b \in R$ for all $a, b \in R$.
• Associativity: $(a * b) * c = a * (b * c)$.

3. Distributive Property

For all $a, b, c \in R$, multiplication distributes over addition:

• Left Distributivity: $a * (b + c) = (a * b) + (a * c)$.
• Right Distributivity: $(b + c) * a = (b * a) + (c * a)$.

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Types of Rings

1. Commutative Ring

A ring is commutative if multiplication is commutative:

$$a * b = b * a, \quad \forall a, b \in R$$

Example: The set of integers *(ℤ, +, ) is a commutative ring.

2. Ring with Identity (Unitary Ring)

A ring R has a multiplicative identity if there exists an element 1 such that:

$$a * 1 = 1 * a = a, \quad \forall a \in R$$

Example: *(ℤ, +, ) is a ring with identity, where 1 is the multiplicative identity.

3. Division Ring (Skew Field)

A ring where every nonzero element has a multiplicative inverse but multiplication is not necessarily commutative.
Example: The set of quaternions ℍ.

4. Field

A ring is a field if it is commutative and every nonzero element has a multiplicative inverse.
Example: ℚ (Rational Numbers), ℝ (Real Numbers), ℂ (Complex Numbers).

5. Integral Domain

A commutative ring with identity (1 ≠ 0) that has no zero divisors (i.e., if $ab = 0$, then either $a = 0$ or $b = 0$).
Example: (ℤ, +, *), the ring of integers.

6. Boolean Ring

A ring where every element satisfies $a * a = a$.
Example: The set $\{0,1\}$ with bitwise XOR as addition and bitwise AND as multiplication.

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Examples of Rings

*1. Integers (ℤ, +, )

• Additive group: $(\mathbb{Z}, +)$ is an abelian group.
• Multiplication is associative and distributive.
• Commutative Ring with Identity (1).

2. Polynomials (ℝ[x], +, ×)

• Set of polynomials with real coefficients under normal addition and multiplication.
• Commutative ring with identity $1$.

3. Matrices (Mₙ(ℝ), +, ×)

• The set of $n \times n$ matrices with real entries.
• Noncommutative ring because matrix multiplication is not commutative.

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Difference Between Rings, Groups, and Fields

Property	Group	Ring	Field
Operations	1	2 (+, *)	2 (+, *)
Addition forms an Abelian Group?	Yes	Yes	Yes
Multiplication forms a Group?	No	No (except in division rings)	Yes (for nonzero elements)
Multiplication is Commutative?	N/A	Sometimes	Yes
Has Multiplicative Inverses?	No	No (except in division rings)	Yes (except 0)