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Magazine Chapter 7: Problem 9
In Exercises 1 through 15 determine the field of quotients of the indicated rings if it exists. If it does not exist, explain why. $$ \mathbb{z}_{12} $$
Short Answer
Expert verified
\( \mathbb{Z}_{12} \) is not an integral domain; thus, its field of quotients does not exist.
Step by step solution
01
Understand the Rings and Fields
In order to determine the field of quotients of the ring \( \mathbb{Z}_{12} \), we first need to understand what these structures represent. The ring \( \mathbb{Z}_{12} \) consists of integers modulo 12, meaning its elements are \( \{0, 1, 2, \, \ldots, \ 11\} \), and operations of addition and multiplication are performed modulo 12. A field, on the other hand, is a set equipped with two operations (addition and multiplication) where each non-zero element has a multiplicative inverse.
02
Check the Ring Properties of \( \mathbb{Z}_{12} \)
A field of quotients exists for an integral domain. An integral domain is a commutative ring with no zero divisors other than zero itself, and it contains a multiplicative identity (1). We need to verify if \( \mathbb{Z}_{12} \) meets these criteria by checking for zero divisors. A zero divisor is a non-zero element \( a \) such that there exists a non-zero element \( b \) where \( a \cdot b \equiv 0 \pmod{12} \).
03
Identify Zero Divisors in \( \mathbb{Z}_{12} \)
Begin by identifying elements that, when multiplied with another non-zero element in the ring, give a result of zero modulo 12. For example, \( 3 \times 4 = 12 \equiv 0 \mod 12 \), making 3 and 4 zero divisors. With zero divisors present, \( \mathbb{Z}_{12} \) is not an integral domain.
04
Conclude About Field of Quotients
Since \( \mathbb{Z}_{12} \) is not an integral domain (it has zero divisors), it does not have a field of quotients. The field of quotients only exists for rings that are integral domains. Therefore, for \( \mathbb{Z}_{12} \), a field of quotients does not exist.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integral Domain
An integral domain is a key concept in understanding when a field of quotients can be formed from a ring. It is a special type of ring that has two main properties:
Without being an integral domain, a ring cannot have a field of quotients.
This is why not all rings can be associated with a field.
- It is commutative with respect to multiplication. This means that changing the order of multiplying two elements does not change the result.
- It has no zero divisors other than zero itself.
Without being an integral domain, a ring cannot have a field of quotients.
This is why not all rings can be associated with a field.
Zero Divisors
Zero divisors are elements within a ring that can multiply with another element to produce zero. In rings like \( \mathbb{Z}_{12} \), they indicate that the ring lacks certain properties such as the ability to form a field of quotients.
Identifying zero divisors is crucial because:
This property confirms that \( \mathbb{Z}_{12} \) contains elements that inhibit forming a field.
Identifying zero divisors is crucial because:
- If a ring has zero divisors, it cannot be an integral domain.
- Zero divisors prevent the ring from having a multiplicative inverse for every non-zero element.
This property confirms that \( \mathbb{Z}_{12} \) contains elements that inhibit forming a field.
Rings
The concept of rings is foundational to understanding structures like fields and integral domains. A ring is a set equipped with two binary operations: addition and multiplication. These operations have to satisfy several properties:
The ring \( \mathbb{Z}_{12} \) is a common example where addition and multiplication are based on modular arithmetic, and not every non-zero element has an inverse.
This characteristic is a distinction between rings and fields.
- Addition must be commutative and associative.
- Multiplication is associative and distributes over addition.
- There is an additive identity (0), and every element has an additive inverse.
The ring \( \mathbb{Z}_{12} \) is a common example where addition and multiplication are based on modular arithmetic, and not every non-zero element has an inverse.
This characteristic is a distinction between rings and fields.
Modular Arithmetic
Modular arithmetic involves calculating numbers while wrapping around upon reaching a certain value, known as the modulus. It is often described informally as "clock arithmetic." For example, the arithmetic used in \( \mathbb{Z}_{12} \) involves numbers from 0 to 11, and calculations are performed modulo 12.
This means:
This property is crucial when determining if a field of quotients can exist for a particular ring.
This means:
- After reaching 11, the next number resets back to 0.
- Operations like addition and multiplication are constrained by this wrapping behavior.
This property is crucial when determining if a field of quotients can exist for a particular ring.
