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Magazine Chapter 19: Problem 1
Let \(A\) be a commutative ring with unity, and \(J\) an ideal of \(A .\) Prove each of the following. 1 \(A / J\) is a commutative ring with unity.
Short Answer
Expert verified
\( A/J \) is a commutative ring with unity, derived from the structure of \( A \) and properties of the ideal \( J \).
Step by step solution
01
Identify Given Information
We know that \( A \) is a commutative ring with unity, meaning it has an addition and multiplication operation that are associative and commutative, there's a multiplicative identity (unity), and every element has an additive inverse. We also know that \( J \) is an ideal of \( A \). An ideal is a subset of \( A \) that is closed under addition and multiplies any element of \( A \) by any element of \( J \) to remain in \( J \).
02
Define the Structure of \( A/J \)
The quotient ring \( A/J \) consists of the set of equivalence classes \([a] = \{a + j | j \in J\} \) for each \( a \in A \). The operations are defined as \([a] + [b] = [a + b] \) and \([a][b] = [ab] \).
03
Check Addition Operation
The addition operation \([a] + [b] = [a + b] \) in \( A/J \) inherits the commutative and associative properties from \( A \) because both operations are defined through the operations in \( A \). There is an additive identity \([0]\) in \( A/J \) since for any \([a]\), \([a] + [0] = [a + 0] = [a]\). Each \([a]\) has an additive inverse \([-a]\) since \([a] + [-a] = [a - a] = [0]\).
04
Check Multiplication Operation
The multiplication operation \([a][b] = [ab]\) is also commutative and associative, again because both are inherited from \( A \).
05
Verify Unity
The unity in \( A/J \) is \([1]\) where \( 1 \) is the multiplicative identity in \( A \). For any \([a] \in A/J\), \([1][a] = [1a] = [a]\) and \([a][1] = [a1] = [a]\). Thus, \([1]\) acts as the multiplicative identity in \( A/J \).
06
Conclusion: \( A/J \) is a Commutative Ring with Unity
Since \( A/J \) satisfies all the properties of a ring and possesses a unity \([1]\) that behaves as a multiplicative identity, \( A/J \) is a commutative ring with unity.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Ring Theory
Ring theory is a fundamental branch of abstract algebra that provides insights into number systems and polynomials. A ring is a set equipped with two binary operations: addition and multiplication. Rings generalize many concepts you may know from integers, such as commutativity, associativity, and distributive laws.
- Commutativity: In a commutative ring, both addition and multiplication are commutative. This means for any two elements \(a\) and \(b\) in the ring, \(a + b = b + a\) and \(a \, b = b \, a\).
- Associativity: Both operations are associative. So for addition, \((a + b) + c = a + (b + c)\) and for multiplication \((a \, b) \, c = a \, (b \, c)\).
- Distributivity: Multiplication distributes over addition, which means \(a \, (b + c) = a \, b + a \, c\).
Ideals
Ideals are special subsets of rings important in ring theory, functioning somewhat like a factor or divisor. An ideal \(J\) in a ring \(A\) is more than just a simple subset. It is closed under ring addition and multiplication by any element in the ring. Here's what makes an ideal distinct:
- Closure under Addition: If \(a\) and \(b\) are in \(J\), then \(a + b\) is also in \(J\).
- Absorbing Behavior: If \(r\) is any element in \(A\) and \(j\) is in \(J\), then \(r \, j\) is in \(J\). This property allows ideals to 'absorb' elements of the ring.
Quotient Ring
A quotient ring, written as \(A/J\), is formed when a ring \(A\) is divided by an ideal \(J\). In simpler terms, it's about building a new ring from equivalence classes derived from \(A\). Here's how it works:
- Equivalence Classes: Each element \(a\) in \(A\) is associated with a set of elements \([a] = \{a + j \, | \, j \in J\}\), where elements behave similarly under ring operations thanks to the ideal \(J\).
- Operations: Addition and multiplication in the quotient ring \(A/J\) take the form \([a] + [b] = [a + b]\) and \([a][b] = [ab]\). These operations are well-defined because they don't depend on the choice of representatives from each equivalent class.
Additive Identity
In mathematics, the additive identity is the element in a set that leaves other elements unchanged when added to them. For any set with an addition operation, such as a ring, there is an element that satisfies this property. In ring theory, this element is typically denoted as 0. Consider the following:
- In Rings: For any element \(a\) in a commutative ring \(A\), the additive identity 0 satisfies \(a + 0 = a\).
- In Quotient Rings: Within a quotient ring \(A/J\), the set \([0]\) serves as the additive identity. For any equivalence class \([a]\), \([a] + [0] = [a]\).
