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Chapter 7: Problem 12

Show that \(\mathbb{Z} \times\\{0\\}\) is an ideal in \(\mathbb{Z} \times \mathbb{Z}\).

Short Answer

Expert verified
\(\mathbb{Z} \times \{0\}\) is an ideal in \(\mathbb{Z} \times \mathbb{Z}\) by satisfying subgroup and absorption properties.

Step by step solution

01

Understanding Ideals

A subset \(I\) of a ring \(R\) is an ideal if: 1) It is a subgroup of \(R\) under addition. 2) For any element \(a\) in \(I\) and any element \(r\) in \(R\), the product \(ar\) is in \(I\). We need to show that \(I = \mathbb{Z} \times \{0\}\) satisfies these properties in the ring \(R = \mathbb{Z} \times \mathbb{Z}\).
02

Checking Subgroup Criterion

The subset \(I = \mathbb{Z} \times \{0\}\) is \(\{(n,0) | n \in \mathbb{Z} \}\). Check closure under addition: if \((n,0), (m,0) \in I\), then \((n+m,0) \in I\) since \(n+m \in \mathbb{Z}\). Check existence of identity: \((0,0) \in I\). Check existence of inverses: the inverse of \((n,0)\) is \((-n,0)\), which is in \(I\) since \(-n \in \mathbb{Z}\). Thus, \(I\) is a subgroup.
03

Checking Absorption Criterion

For \((n,0) \in I\) and \((a,b) \in R\), calculate \((n,0) \cdot (a,b) = (na,0b) = (na,0)\). Since \(na \in \mathbb{Z}\), \((na,0) \in I\). This shows any element in \(I\) when multiplied by any element in \(R\) is still in \(I\). Thus, \(I\) satisfies the absorption property of ideals.
04

Conclusion

Since \(I = \mathbb{Z} \times \{0\}\) satisfies both the subgroup and absorption criteria, it is indeed an ideal of the ring \(\mathbb{Z} \times \mathbb{Z}\).

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Key Concepts

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

Subgroup Criterion
To determine if a subset is a subgroup, we use the subgroup criterion, which needs the subset to be a subgroup under addition. This involves checking:
  • Closure under addition: When we add any two elements from the subset, the result must also be an element of the subset. In our example, if \((n,0)\) and \((m,0)\) are in \(\mathbb{Z} \times \{0\}\), then \((n+m,0)\) is also in \(\mathbb{Z} \times \{0\}\).

  • Existence of identity element: The identity element in \(\mathbb{Z} \times \{0\}\) for addition is \((0,0)\), which is present as \((0,0)\) belongs to the subset.

  • Existence of inverses: For every element \((n,0)\) in the subset, the inverse element \((-n,0)\) (which cancels the original element when added) should also be in the subset. Since `-n` is in \(\mathbb{Z}\), \((-n,0)\) is indeed in \(\mathbb{Z} \times \{0\}\).
Thus, ensuring these properties proves \(\mathbb{Z} \times \{0\}\) is a subgroup of \(\mathbb{Z} \times \mathbb{Z}\).
Absorption Property
The absorption property is crucial to understanding how a subset can stay within itself even when interacting with the larger structure of the ring. Specifically, for ideals, it means:
  • When an element from the subset is multiplied by any element of the ring, the result should still be an element of the subset. This is a vital trait for an ideal.

In the case of \(\mathbb{Z} \times \{0\}\), consider any \((n,0)\) from \(\mathbb{Z} \times \{0\}\) and any \((a,b)\) from the ring \(\mathbb{Z} \times \mathbb{Z}\). Their product is \((n,0) \cdot (a,b) = (na,0b) = (na,0)\). Since \(na\) is in \(\mathbb{Z}\), \((na,0)\) remains an element of \(\mathbb{Z} \times \{0\}\). This shows that \(\mathbb{Z} \times \{0\}\) observes the absorption property effectively, which is why it qualifies as an ideal.
Ring Theory
Ring theory is a fundamental aspect of modern algebra that involves the study of sets (called rings) equipped with two operations commonly known as addition and multiplication. In this area:
  • A ring is a set equipped with two binary operations that generalize addition and multiplication.

  • The set must satisfy several properties, such as closure, associativity, distributivity of multiplication over addition, and the existence of a multiplicative identity.

  • Rings can include familiar structures such as integers \((\mathbb{Z})\), and they extend complex algebraic constructs.
In our example, \(\mathbb{Z} \times \mathbb{Z}\) is a ring, meaning it's structured in a way to accommodate these operations fully. Understanding ring theory helps recognize that certain substructures, like ideals, fulfill particular roles which maintain the properties of the ring even when breaking down into smaller parts.
Closure under Addition
Closure under addition is a crucial feature for any subset to be a subgroup within a ring. This property ensures stability within the subset when performing addition operations:
  • Any two elements from the subset, when added, result in another element still within the same subset.

In \(\mathbb{Z} \times \{0\}\), if you take any two elements \((n,0)\) and \((m,0)\), their sum \((n+m,0)\) also lies in \(\mathbb{Z} \times \{0\}\). The subset closes on itself under addition, ensuring it remains part of the larger ring \(\mathbb{Z} \times \mathbb{Z}\) with this operation. Achieving closure is a fundamental step in verifying subgroup properties, making it essential for establishing whether a given subset qualifies.

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