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Rings and Fields

  • Understand the definition and properties of rings.
  • Learn about fields and their characteristics.
  • Explore examples of rings and fields.

Definition of Rings

A ring is a set \( R \) equipped with two binary operations: addition (+) and multiplication (·) that satisfy the following properties:

  • \((R, +)\) is an abelian group.
  • \((R, \cdot)\) is a monoid.
  • Multiplication is distributive over addition: for all \(a, b, c \in R\), \(a \cdot (b + c) = a \cdot b + a \cdot c\) and \((a + b) \cdot c = a \cdot c + b \cdot c\).

Properties of Rings

  • Associativity of Addition and Multiplication: Addition and multiplication are associative operations.
  • Commutativity of Addition: Addition is commutative.
  • Distributive Property: Multiplication is distributive over addition.
  • Existence of Additive Identity: There exists an additive identity (0) such that for all \(a \in R\), \(a + 0 = a\).
  • Existence of Additive Inverses: For every \(a \in R\), there exists an element \(-a \in R\) such that \(a + (-a) = 0\).

Definition of Fields

A field is a ring \( F \) in which every non-zero element has a multiplicative inverse. Formally, a field is a set \( F \) equipped with two operations, addition (+) and multiplication (·), such that:

  • \((F, +)\) is an abelian group.
  • \((F \setminus \{0\}, \cdot)\) is an abelian group.
  • Multiplication is distributive over addition.

Properties of Fields

  • Commutativity of Multiplication: Multiplication is commutative.
  • Existence of Multiplicative Identity: There exists a multiplicative identity (1) such that for all \(a \in F\), \(a \cdot 1 = a\).
  • Existence of Multiplicative Inverses: For every non-zero element \(a \in F\), there exists an element \(a^{-1} \in F\) such that \(a \cdot a^{-1} = 1\).

Examples of Rings and Fields

  • Example 1: The set of integers \(\mathbb{Z}\) with the usual addition and multiplication is a ring.
  • Example 2: The set of rational numbers \(\mathbb{Q}\) with the usual addition and multiplication is a field.
  • Example 3: The set of polynomials with real coefficients, \(\mathbb{R}[x]\), is a ring.
  • Example 4: The set of real numbers \(\mathbb{R}\) and the set of complex numbers \(\mathbb{C}\) with the usual addition and multiplication are fields.

Exercises

  • Identify Rings: Determine if the set of \(2 \times 2\) matrices with real entries forms a ring with matrix addition and multiplication.
  • Identify Fields: Verify if the set of complex numbers \(\mathbb{C}\) forms a field.
  • Ring Homomorphisms: Show that the function \(\phi: \mathbb{Z} \to \mathbb{Z}_6\) defined by \(\phi(a) = a \mod 6\) is a ring homomorphism.

Summary

In this lesson, we explored the concepts of rings and fields, including their definitions, properties, and examples. Understanding these fundamental ideas is crucial as they form the building blocks for more advanced topics in abstract algebra and number theory.

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