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Magazine Chapter 7: Problem 20
Let \(R_{1}\) and \(R_{2}\) be rings. (a) Let \(\phi: R_{1} \times R_{2} \rightarrow R_{1}\) be defined by \(\phi(a, b)=a\). Show that \(\phi\) is a ring homomorphism. (b) Show that \(R_{1} \times R_{2} \cong R_{2} \times R_{1}\)
Short Answer
Expert verified
(a) \( \phi \) is a ring homomorphism. (b) \( R_{1} \times R_{2} \cong R_{2} \times R_{1} \).
Step by step solution
01
Understanding Ring Homomorphisms
A ring homomorphism is a function between two rings that respects the ring operations (addition and multiplication) and maps the identity element of the first ring to the identity element of the second ring. For a function \( \phi: R_{1} \times R_{2} \rightarrow R_{1} \), it must hold that for all \( (a,b), (c,d) \in R_{1} \times R_{2} \):\[ \phi((a,b)+(c,d)) = \phi(a+c, b+d) = a+c = \phi(a,b) + \phi(c,d) \] and \[ \phi((a,b)(c,d)) = \phi(ac, bd) = ac = \phi(a,b)\phi(c,d) \]. Additionally, it must map the identity: \( \phi((1,0)) = 1 \).
02
Proving Addition Preservation
First, check if \( \phi \) preserves addition. We need to show that: \( \phi((a,b) + (c,d)) = \phi(a,b) + \phi(c,d) \). Calculate: \[ \phi((a,b) + (c,d)) = \phi((a+c, b+d)) = a+c. \] Now calculate: \( \phi(a,b) + \phi(c,d) = a + c \). Both equal \( a+c \), hence, addition is preserved.
03
Proving Multiplication Preservation
Next, we need to show that \( \phi \) preserves multiplication. Check that: \( \phi((a,b)(c,d)) = \phi(a,b)\phi(c,d) \). Compute: \[ \phi((a,b)(c,d)) = \phi((ac, bd)) = ac. \] Since: \( \phi(a,b)\phi(c,d) = a \cdot c \), both are equal, indicating multiplication is preserved.
04
Confirming Identity Mapping
Verify that \( \phi \) maps the identity of \( R_{1} \times R_{2} \) to the identity of \( R_{1} \): \( \phi((1,0)) = 1 \). This is straightforward as \( \phi(1,0) = 1 \).
05
Showing Conjugacy Between Product Rings
To show \( R_{1} \times R_{2} \cong R_{2} \times R_{1} \), we need a bijective ring homomorphism. Define \( \psi: R_{1} \times R_{2} \rightarrow R_{2} \times R_{1} \) by \( \psi(a,b) = (b,a) \). This swap is clearly bijective and maintains both operations:1. Addition: \( \psi((a,b) + (c,d)) = \psi((a+c, b+d)) = (b+d, a+c) \) and \( \psi(a,b) + \psi(c,d) = (b,a) + (d,c) = (b+d, a+c) \).2. Multiplication: \( \psi((a,b)(c,d)) = \psi((ac, bd)) = (bd, ac) \), \( \psi(a,b)\psi(c,d) = (b,a)(d,c) = (bd, ac) \).
06
Confirm Bijectiveness of \( \psi \)
Verify that \( \psi \) is bijective:- **Injective**: If \( \psi(a,b) = \psi(c,d) \Rightarrow (b,a) = (d,c) \), implying \( a = c \) and \( b = d \).- **Surjective**: For every element \((b,a)\) in \( R_{2} \times R_{1} \), there exists an \((a,b)\) in \( R_{1} \times R_{2} \) such that \( \psi(a,b) = (b,a) \). Thus, \( \psi \) is a bijection.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Rings
A ring is a mathematical structure that is defined by two operations: addition and multiplication. These operations must satisfy certain properties for a set to be considered a ring. Here are the basic properties of rings:
- Closure under Addition and Multiplication: The sum and product of any two elements in the ring should also be in the ring.
- Associative Property: Both addition and multiplication are associative in rings, meaning \((a + b) + c = a + (b + c)\) and \((ab)c = a(bc)\) hold for any elements \(a, b,\) and \(c\) in the ring.
- Commutative Addition: Addition in rings is always commutative: \(a + b = b + a\).
- Additive Identity: There exists an element, commonly denoted by \(0\), such that adding it to any element \(a\) in the ring leaves \(a\) unchanged: \(a+0 = a\).
- Additive Inverses: For every element \(a\), there is an element \(-a\) such that \(a + (-a) = 0\).
Ring Isomorphism
Ring isomorphism is a concept that indicates a structural similarity between two rings. When two rings \(R_1\) and \(R_2\) are isomorphic, it means there is a bijective function between them that not only pairs elements uniquely but also respects both the addition and multiplication operations. This function is termed a ring isomorphism.To show that two rings are isomorphic, one must:
- Find a bijective (one-to-one and onto) function \( \psi: R_1 \rightarrow R_2 \).
- Prove that \( \psi \) respects addition, meaning \( \psi(a + b) = \psi(a) + \psi(b)\) for any \(a, b\) in \(R_1\).
- Show that \( \psi \) maintains multiplication: \( \psi(ab) = \psi(a) \cdot \psi(b)\) for any \(a, b\) in \(R_1\).
Ring Operations
Ring operations primarily revolve around addition and multiplication, which are fundamental to the structure of any ring.**Addition:**
- Addition in rings is always associative and commutative, so you can rearrange and group added elements without changing the result.
- Every ring must have an additive identity, \(0\), where \(a + 0 = a\) for any element \(a\) in the ring.
- Each element must also have an additive inverse, meaning for every element \(a\), there is a \(-a\) such that \(a + (-a) = 0\).
- Multiplication in rings needs to be associative, such that \((ab)c = a(bc)\) for any elements \(a, b,\) and \(c\).
- Not all rings require multiplication to be commutative, but it is a common property in rings like the integers.
- Rings with unity possess a multiplicative identity, \(1\), where \(a \cdot 1 = a\).
