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Chapter 18: Problem 25

Describe all ring homomorphisms of \(\mathbb{Z} \times \mathbb{Z}\) into \(Z\).

Short Answer

Expert verified
Ring homomorphisms are of the form \( \phi((a, b)) = ma + nb \) with \( m + n = 1 \). Possible mappings include \( (a, b) \rightarrow a \) or \( (a, b) \rightarrow b \).

Step by step solution

01

Understand the Problem

We are tasked with describing all possible ring homomorphisms from the ring \( \mathbb{Z} \times \mathbb{Z} \) into the ring \( \mathbb{Z} \). A ring homomorphism should preserve addition and multiplication and map the multiplicative identity to the multiplicative identity.
02

Properties of \( \mathbb{Z} \times \mathbb{Z} \) and \( \mathbb{Z} \)

The ring \( \mathbb{Z} \times \mathbb{Z} \) consists of ordered pairs \((a, b)\) where both \(a\) and \(b\) are integers. The ring \( \mathbb{Z} \) consists of usual integers. Note that 1 in \( \mathbb{Z} \times \mathbb{Z} \) is \((1, 1)\) and 1 in \( \mathbb{Z} \) is just 1.
03

Define General Form of a Homomorphism

For a ring homomorphism \( \phi: \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z} \), we express it as \( \phi((a, b)) = f(a, b) \) for some function \( f \). Since it's a ring homomorphism, it should preserve addition (i.e. \( \phi((a_1 + a_2, b_1 + b_2)) = \phi((a_1, b_1) + (a_2, b_2)) \)), multiplication, and send the element \((1, 1)\) to 1.
04

Check Homomorphic Conditions

The condition \( \phi((a, 0)) = a\phi((1,0)) \) and \( \phi((0, b)) = b\phi((0,1)) \) must hold. Consider specific mappings utilizing these observations. Common forms of \( \phi((a, b)) \) can be expressed as \( ma + nb \) for some integers \( m \) and \( n \), often involving placeholder elements like \( \phi((1, 0)) \) and \( \phi((0, 1)) \).
05

Verify Identity Mapping

The multiplicative identity \((1, 1)\) in \( \mathbb{Z} \times \mathbb{Z} \) must map to the multiplicative identity 1 in \( \mathbb{Z} \). Hence, we require \( m + n = 1 \).
06

Conclude the Homomorphisms

The ring homomorphisms from \( \mathbb{Z} \times \mathbb{Z} \) to \( \mathbb{Z} \) are functions of the form \( \phi((a, b)) = ma + nb \) where \( m + n = 1 \). Therefore, the possible mappings are \( \phi((a, b)) = a \) or \( \phi((a, b)) = b \) or integer affine combinations that satisfy the condition.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Properties of Rings
Rings are fundamental structures in algebra and have specific properties. A ring is a set equipped with two binary operations: addition and multiplication. To qualify as a ring, these operations must follow certain rules:
  • Addition must be commutative: For any elements \( a \) and \( b \) in the ring, \( a + b = b + a \).
  • Additive identity: There exists an element \( 0 \), such that \( a + 0 = a \) for any element \( a \) in the ring.
  • Multiplication is associative: For any elements \( a, b, \) and \( c \), \( (a \, b) \, c = a \, (b \, c) \).
  • Additive inverse: For every \( a \), there is an element \( -a \), such that \( a + (-a) = 0 \).
  • Distributive properties: Multiplication distributes over addition, meaning \( a \, (b + c) = a \, b + a \, c \) and \((b + c) \, a = b \, a + c \, a \).
These properties ensure the behavior of elements in a ring stays consistent and forms the basis for further algebraic operations and structures.
Understanding these basics aids in comprehending more complex concepts, such as ring homomorphisms, where these properties are preserved during mappings between rings.
Ring Homomorphisms
A ring homomorphism is a function between two rings that respects the operations of addition and multiplication, along with the identity elements. Here are the essential characteristics:
  • Preservation of addition: If \( \phi \) is a homomorphism, then for any elements \( a \) and \( b \) in the first ring, \( \phi(a + b) = \phi(a) + \phi(b) \).
  • Preservation of multiplication: Similarly, \( \phi(a \, b) = \phi(a) \, \phi(b) \).
  • Identity mapping: The multiplicative identity element in the source ring must map to the multiplicative identity in the target ring: \( \phi(1) = 1 \).
These properties ensure that a ring homomorphism maintains the structure of the rings involved. In the case of mapping from \( \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z} \), the function \( \phi \) would be crafted to respect these conditions.
By focusing on these conditions, you can determine possible functions that constitute valid homomorphisms, often involving specific integer linear combinations as seen in the solution steps.
Integers as a Ring
The integers \( \mathbb{Z} \) represent one of the simplest examples of a ring, often considered a commutative ring with unity. Here, basic ring properties hold:
  • Closure under addition and multiplication: The sum or product of any two integers is an integer.
  • Associative and commutative properties: Both addition and multiplication are associative and commutative.
  • Additive identity: The number 0 serves as the additive identity, satisfying \( a + 0 = a \).
  • Multiplicative identity: The number 1 serves as the multiplicative identity.
  • Additive inverses: For every integer \( a \), there exists \( -a \) such that \( a + (-a) = 0 \).
When considering the mapping to the ring of integers, it's crucial to maintain these properties through homomorphism.
Because the integers form a ring with these well-defined characteristics, they provide a foundational model for understanding more complex algebraic structures and demonstrate core examples of ring homomorphisms in action, as seen in mappings from \( \mathbb{Z} \times \mathbb{Z} \).

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Describe all ring homomorphisms of \(Z \times Z\) into \(\mathbb{Z}\).

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