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Ring and Basic Properties

Definition of a Ring

A ring is a set \( R \) equipped with two binary operations: addition (+) and multiplication (\( \cdot \)) such that:

  1. \( (R, +) \) is an abelian group (i.e., it satisfies closure, associativity, identity, inverses, and commutativity under addition).
  2. \( (R, \cdot) \) is a semigroup (i.e., it satisfies closure and associativity under multiplication).
  3. Distributive properties hold:
    • \( a \cdot (b + c) = (a \cdot b) + (a \cdot c) \)
    • \( (a + b) \cdot c = (a \cdot c) + (b \cdot c) \)

Examples

  1. \( \mathbb{Z} \): Integers under addition and multiplication form a ring.
  2. \( \mathbb{R}[x] \): Polynomials with real coefficients form a ring under standard addition and multiplication.

Non-Examples

  1. Natural numbers \( \mathbb{N} \) under addition and multiplication are not a ring because they lack additive inverses.

Properties of Rings

Types of Rings

Commutative Ring

A ring \( R \) is commutative if multiplication is commutative:

\[ \forall a, b \in R, \quad a \cdot b = b \cdot a \]

Ring with Unity

A ring \( R \) has a multiplicative identity (unity) if there exists \( 1 \in R \) such that:

\[ 1 \cdot a = a \cdot 1 = a \quad \forall a \in R \]

Division Ring

A ring \( R \) is a division ring if every nonzero element has a multiplicative inverse.

Zero Divisors

An element \( a \in R \) is a zero divisor if:

\[ \exists b \neq 0 \in R, \quad a \cdot b = 0 \quad \text{or} \quad b \cdot a = 0 \]

Integral Domain

A commutative ring \( R \) with no zero divisors is an integral domain.

Example: Verify Properties

  • Set: \( \mathbb{Z}_6 \) under addition and multiplication modulo 6.
  • Check if it satisfies the ring properties.

Solution

  1. Closure: \( a + b \mod 6 \) and \( a \cdot b \mod 6 \) remain in \( \mathbb{Z}_6 \).
  2. Associativity: Both addition and multiplication are associative.
  3. Distributivity: Holds by modular arithmetic rules.
  4. Zero divisors: \( 2 \cdot 3 \mod 6 = 0 \).
  5. Hence, \( \mathbb{Z}_6 \) is not an integral domain.

Field

Definition of a Field

A field is a commutative ring \( F \) with unity where every nonzero element has a multiplicative inverse:

\[ \forall a \in F, a \neq 0, \quad \exists b \in F \text{ such that } a \cdot b = 1 \]

Examples

  1. \( \mathbb{Q} \): Rational numbers under standard addition and multiplication.
  2. \( \mathbb{R} \): Real numbers under standard addition and multiplication.
  3. \( \mathbb{Z}_p \): Integers modulo a prime \( p \), where addition and multiplication modulo \( p \) form a field.

Non-Examples

  1. \( \mathbb{Z} \): Integers are not a field because most elements lack multiplicative inverses.

Field Properties

  1. Addition and multiplication are commutative.
  2. Distributive property holds.
  3. Every nonzero element has a multiplicative inverse.

Applications

  1. Cryptography: Fields like \( \mathbb{Z}_p \) are used in RSA and elliptic curve cryptography.
  2. Linear Algebra: Vector spaces require a field for scalar multiplication.

Tips and Tricks

  1. Check Zero Divisors: To determine if a ring is an integral domain, verify there are no zero divisors.
  2. Prime Modulo: For fields \( \mathbb{Z}_p \), ensure \( p \) is prime.
  3. Unity Element: Always confirm the presence of a multiplicative identity when analyzing rings and fields.
  4. Inverse Element: In fields, test the existence of multiplicative inverses for all nonzero elements.