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Ring Homomorphisms and Ideals

  • Understand the definition and properties of ring homomorphisms.
  • Learn about ideals and their characteristics.
  • Explore examples of ring homomorphisms and ideals.

Definition of Ring Homomorphisms

A ring homomorphism is a function between two rings that preserves both the addition and multiplication operations. If \( R \) and \( S \) are rings, a function \( \phi: R \to S \) is a homomorphism if for all \( a, b \in R \):

  • \( \phi(a + b) = \phi(a) + \phi(b) \)
  • \( \phi(a \cdot b) = \phi(a) \cdot \phi(b) \)

Properties of Ring Homomorphisms

  • Identity Preservation: If \( 0 \) is the additive identity in \( R \), then \( \phi(0) \) is the additive identity in \( S \).
  • Preservation of Unity (if applicable): If \( R \) and \( S \) have multiplicative identities \( 1_R \) and \( 1_S \) respectively, then \( \phi(1_R) = 1_S \).
  • Kernel of a Homomorphism: The kernel of a ring homomorphism \( \phi: R \to S \) is the set of elements in \( R \) that map to the additive identity in \( S \), denoted as \( \ker(\phi) = \{r \in R \mid \phi(r) = 0\} \).

Definition of Ideals

An ideal is a special subset of a ring that absorbs multiplication by elements of the ring. Formally, a subset \( I \) of a ring \( R \) is an ideal if:

  • \( (I, +) \) is a subgroup of \( (R, +) \).
  • For all \( r \in R \) and \( i \in I \), both \( r \cdot i \) and \( i \cdot r \) are in \( I \).

Properties of Ideals

  • Absorption by Multiplication: If \( I \) is an ideal in \( R \), then for every \( r \in R \) and \( i \in I \), \( r \cdot i \in I \) and \( i \cdot r \in I \).
  • Sum and Intersection: The sum and intersection of two ideals in a ring are also ideals.
  • Principal Ideals: An ideal generated by a single element \( a \in R \) is called a principal ideal, denoted as \( (a) = \{r \cdot a \mid r \in R\} \).

Examples of Ring Homomorphisms and Ideals

  • Example 1: Consider the rings \( (\mathbb{Z}, +, \cdot) \) and \( (\mathbb{Z}_n, +, \cdot) \). The function \( \phi: \mathbb{Z} \to \mathbb{Z}_n \) defined by \( \phi(a) = a \mod n \) is a ring homomorphism.
  • Example 2: The set of even integers \( 2\mathbb{Z} \) is an ideal in the ring \( \mathbb{Z} \).
  • Example 3: The kernel of the ring homomorphism \( \phi: \mathbb{Z} \to \mathbb{Z}_6 \) defined by \( \phi(a) = a \mod 6 \) is the ideal \( 6\mathbb{Z} \) in \( \mathbb{Z} \).

Exercises

  • Identify Ring Homomorphisms: Given the rings \( \mathbb{R}[x] \) and \( \mathbb{C} \), determine if the function \( \phi: \mathbb{R}[x] \to \mathbb{C} \) defined by \( \phi(f(x)) = f(i) \) is a ring homomorphism, where \( i \) is the imaginary unit.
  • Identify Ideals: Verify if the set of polynomials in \( \mathbb{R}[x] \) that are divisible by \( x^2 + 1 \) forms an ideal.
  • Kernel and Image: Find the kernel and image of the ring homomorphism \( \phi: \mathbb{Z} \to \mathbb{Z}_{10} \) defined by \( \phi(a) = a \mod 10 \).

Summary

In this lesson, we explored the concepts of ring homomorphisms and ideals, including their definitions, properties, and examples. Understanding these fundamental ideas is crucial as they provide insights into the structure of rings and their substructures.

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