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Chapter 18: Problem 14

In Exercises 14 through 19 , describe all units in the given ring \(\mathbf{z}\)

Short Answer

Expert verified
Units in \(\mathbb{Z}\) are \(+1\) and \(-1\).

Step by step solution

01

Understanding Units

In a ring, a unit is an element that has a multiplicative inverse within the same set. This means if \(a\) is a unit in a ring \(R\), there exists some \(b\) in \(R\) such that \(a \cdot b = 1\). We need to identify the units within the ring of integers, \(\mathbb{Z}\).
02

Identifying Units in \(\mathbb{Z}\)

The integers are a specific type of ring. In \(\mathbb{Z}\), the only integers that have multiplicative inverses are \(1\) and \(-1\). This is because neither \(0\) nor any integer other than \(\pm 1\) has a multiplicative inverse in \(\mathbb{Z}\).
03

Conclusion

In the ring of integers \(\mathbb{Z}\), the units are specifically \(+1\) and \(-1\), because these are the only numbers that when multiplied by their respective opposite \(+1\) or \(-1\) yield the multiplicative identity, which is \(1\). Any other integer would not satisfy this requirement.

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Key Concepts

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

Units in a ring
In the realm of ring theory, understanding what a "unit" is can help us grasp how elements interact within a ring. A ring is a set equipped with two operations, addition and multiplication, that behave in particular ways. A unit in a ring is a special element. It has a partner, or a multiplicative inverse, such that when they are multiplied together, they result in the multiplicative identity of the ring, typically denoted as 1.
For example:
  • If we have a unit \( a \) in a ring \( R \), there will be an element \( b \) also in \( R \)
  • They should satisfy the equation \( a \cdot b = 1 \)
This property of possessing a multiplicative inverse is what transforms an element from just being in a ring to being a unit in that ring. Understanding units is crucial for solving equations within rings.
Multiplicative inverse
The concept of a multiplicative inverse is closely tied to that of a unit. Think of it as having an "undo" button for multiplication. If you multiply a number by its multiplicative inverse, you end up with the neutral or identity element, which is 1 in most rings.In simpler terms, if \( a \) is a number and \( a^{-1} \) is its multiplicative inverse, the equation \( a \cdot a^{-1} = 1 \) holds true.
Some key points about multiplicative inverses:
  • Only certain elements in a given mathematical set will have a multiplicative inverse.
  • In the set of real numbers, every non-zero number has an inverse.
  • In a general ring, identifying which elements have inverses can be more complex.
Understanding multiplicative inverses is central when exploring rings as it underpins more complex operations and equations.
Integers as a ring
The integers, represented by \( \mathbb{Z} \), provide an easy way to illustrate the concepts of ring theory. As a ring, \( \mathbb{Z} \) is equipped with the standard operations of addition and multiplication.In the ring of integers:
  • The only units are 1 and -1.
  • This is because 1 multiplied by 1 gives 1, and -1 multiplied by -1 also gives 1, fulfilling the condition to be considered a unit.
  • Elements like 2 or -3 don't have such an inverse within \( \mathbb{Z} \) because they can't produce the identity element when multiplied by any other integer.
One can see \( \mathbb{Z} \) as a simple playground to understand broader concepts in ring theory, where the conditions and restrictions of units become more evident.

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Most popular questions from this chapter

(Freshman exponentiation) Let \(p\) be a prime. Show that in the ring \(Z_{p}\) we have \((a+b)^{p}=a^{p}+b^{p}\) for all \(a, b \in \mathbb{Z}_{p}\). [Hint: Observe that the usual binomial expansion for \((a+b)^{n}\) is valid in a comnutative ring.]

In Exercises 7 through 13, 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. \(\mathbb{Z} \times \mathbb{Z}\) with addition and multiplication by components

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

In Exercises 7 through 13, 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. \((a+b \sqrt{2} \mid a, b \in \mathbb{Z}\\}\) with the usual addition and multiplication

Show that the unity element in a subfield of a field must be the unity of the whole field, in contrast to Exercise 32 for rings.
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