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Chapter 18: Problem 6

Compute the product in the given ring. $$ (-3,5)(2,-4) \text { in } Z_{4} \times z_{11} $$

Short Answer

Expert verified
The product is \((2, 2)\) in the ring \( \mathbb{Z}_4 \times \mathbb{Z}_{11} \).

Step by step solution

01

Understand the Ring

The given problem involves the ring \( \mathbb{Z}_4 \times \mathbb{Z}_{11} \). This means operations for each component of the ordered pair are performed under modulo 4 and modulo 11 respectively.
02

Multiply the First Components

Take the first components from each pair, \(-3\) and \(2\), and calculate their product in \( \mathbb{Z}_4 \). The multiplication of \(-3 \times 2 = -6\). Next, compute \(-6 \mod 4\). Since \(-6 \mod 4 = 2\), the result of the first component is \(2\).
03

Multiply the Second Components

Take the second components from each pair, \(5\) and \(-4\), and calculate their product in \( \mathbb{Z}_{11} \). The multiplication of \(5 \times (-4) = -20\). Compute \(-20 \mod 11\). Since \(-20 \equiv 2 \pmod{11}\), the result for the second component is \(2\).
04

Combine Results

Combine the results of the two components obtained. The new pair in the ring \( \mathbb{Z}_4 \times \mathbb{Z}_{11} \) is \((2, 2)\).

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Key Concepts

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

modular arithmetic
Modular arithmetic might seem confusing at first, but it's simply a system used for integers where numbers "wrap around" when reaching a certain value. This value is called the modulus. Similar to how a clock resets after reaching 12, modular arithmetic resets its count after reaching the modulus.

For example, in modular arithmetic with a modulus of 4 (denoted as \(\mathbb{Z}_4\)), numbers are reduced to their remainder when divided by 4:
  • 7 mod 4 = 3 (since 4 goes into 7 once with a remainder of 3)
  • 10 mod 4 = 2 (since 4 goes into 10 twice with a remainder of 2)

This method is especially useful in operations within specific sets, such as when you perform multiplications or additions with numbers under certain constraints, like in structured rings and fields.
direct product of rings
The direct product of rings involves combining two or more rings into a larger one, where operations are defined component-wise. Imagine two rings, \(\mathbb{Z}_4\) and \(\mathbb{Z}_{11}\). Each pair of elements, like \((a, b)\) in these rings, operate independently under their respective modulus.

When computing in the direct product, you:
  • Apply operations like addition and multiplication separately within their own ring.
  • Use modular arithmetic to handle each component according to its modulus.

In our computation example, we multiply and adjust \(-3\) and \(2\) using mod 4, and \(5\) and \(-4\) using mod 11, treating each part of the ordered pair distinctly while computing the overall result.
integer modulo
The integer modulo operation is a fundamental aspect of modular arithmetic and involves finding the remainder after division of one number by another. Notated as \(a \mod n\), where \(a\) is the integer to be divided and \(n\) is the modulus.

For example:
  • \(-20 \mod 11\) means determining what remains when \(-20\) is divided by 11.
  • \(-20\) divided by 11 leaves a remainder of 2, so \(-20 \equiv 2 \pmod{11}\).

Understanding integer modulos allows you to simplify calculations, especially when numbers might be negative or exceed the modulus, making it indispensable in ring theory to keep results within specified limits.

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Most popular questions from this chapter

In Exercises 1 through 6 , compute the product in the given ring. (12) (16) in \(Z_{24}\)

Show that a ring \(R\) has no nonzero nilpotent element if and only if 0 is the only solution of \(x^{2}=0\) in \(R\).

Decide whether the indicated operations of addition and multiplication are defined (closed) on the set, and give a ring structure. If a ring is not formed, tell why this is the case. If a ring is formed, state whether the ring is commutative, whether it has unity, and whether it is a field. $$ \mathrm{Z} \text { " with the usual addition and multiplication. } $$

Correct the definition of the italicized term without reference to the text, if correction is needed, so that it is in a from acceptable for publication. A field \(F\) is a ring with nonzero unity such that the set of nonzero elements of \(F\) is a group under multiplication.

In Exercises 1 through 6 , compute the product in the given ring. \((16)(3)\) in \(Z_{12}\)
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