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Chapter 14: Problem 5

If \(a\) is a unit in ring \(R\), prove that \(-a\) is also a unit in \(R\).

Short Answer

Expert verified
Given that the unit \(a\) in ring \(R\) has \(b\) as its multiplicative inverse, the proof showed that \(-a\) is also a unit in \(R\), with \(-b\) as its multiplicative inverse. Thus, it's confirmed that if \(a\) is a unit in \(R\), then \(-a\) is also a unit in \(R\).

Step by step solution

01

Understanding the Meaning of Unit in a Ring

Start by understanding that a unit in a ring \(R\) is an element that has a multiplicative inverse. This means that if \(a\) is a unit in ring \(R\), there exists an element \(b\) in \(R\) such that their product (\(a*b\) or \(b*a\)) equals the multiplicative identity of \(R\), which is usually denoted as 1.
02

Finding the Multiplicative Inverse of -a

To prove that \(-a\) is a unit in \(R\), you need to find an element \(b'\) so that multiplying \(-a\) by \(b'\) yields the multiplicative identity, 1. To find \(b'\), consider \(-b\) (the additive inverse of \(b\)). Now verify the product \((-a)*(-b)\).
03

Using Ring Properties to Simplify the Expression

You can simplify the expression \(-a*-b\) by using the properties of the ring. By distributive property, \(-a*-b = -(a*-b) = -1\). Now due to the existence of additive inverse in the ring, \(-1 = 1\). So, \(-a*-b = 1.\)
04

Concluding the Proof

You've found that \(-a\) and \(-b\) multiply to give 1, which defines the requirement for a unit in the ring \(R\). This confirms that if \(a\) is a unit in \(R\), with \(b\) as its multiplicative inverse, \(-a\) is also a unit in \(R\). And its multiplicative inverse is \(-b\).

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Most popular questions from this chapter

How many units and how many (proper) zero divisors are there in (a) \(\mathbf{Z}_{17}\) ? (b) \(\mathbf{Z}_{117}\) ? (c) \(\mathbf{Z}_{1117}\) ?

a) Let \((R,+, \cdot)\) be a finite commutative ring with unity \(u\). If \(r \in R\) and \(r\) is not the zero element of \(R\), prove that \(r\) is either a unit or a proper divisor of zero. b) Does the result in part (a) remain valid when \(R\) is infinite?

Let \(\left(R,+,^{\circ}\right)\) be a ring with unity \(u\), and \(|R|=8 .\) On \(R^{4}=R \times R \times R \times R\), define \(+\) and - as suggested by Exercise 22 . In the ring \(R^{4}\), (a) how many elements have exactly two nonzero components? (b) how many elements have all nonzero components? (c) is there a unity? (d) how many units are there if \(R\) has four units?

Find the multiplicative inverse of each element in \(\mathbf{Z}_{11}, \mathbf{Z}_{13}\), and \(\mathbf{Z}_{17}\).

Given \(n\) positive integers \(x_{1}, x_{2}, \ldots, x_{n}\), not necessarily distinct, prove that either \(n \mid\left(x_{1}+x_{2}+\cdots+x_{i}\right)\), for some \(1 \leq i \leq n\), or there exist \(1 \leq i
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