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Chapter 14: Problem 14

Give an example of a ring with eight elements. How about one with 16 elements? Generalize.

Short Answer

Expert verified
Examples of rings with eight and sixteen elements are \(Z_8 = \{0, 1, 2, 3, 4, 5, 6, 7\}\) and \(Z_{16} = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15\}\), respectively. Generally, for any positive integer n, a ring with \(2^n\) elements exists and can be represented as \(Z_{2^n}\) under the usual modulo \(2^n\) operations.

Step by step solution

01

Example of a Ring With Eight Elements

One such example can be a ring \(Z_8 = \{0, 1, 2, 3, 4, 5, 6, 7\}\) under the usual operations of addition and multiplication modulo 8.
02

Example of a Ring With Sixteen Elements

An example could be a ring \(Z_{16} = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15\}\) following the regular conventions of addition and multiplication modulo 16.
03

Generalizing the Rule

In general, for any positive integer n, there exists a ring with \(2^n\) elements. A known example can be the ring \(Z_{2^n}\) under the standard modulo \(2^n\) operations.

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Most popular questions from this chapter

Determine whether or not each of the following sets of numbers is a ring under ordinary addition and multiplication. a) \(R=\) the set of positive integers and zero b) \(R=\\{k n \mid n \in \mathbf{Z}, k\) a fixed positive integer\\} c) \(R=\\{a+b \sqrt{2} \mid a, b \in \mathbf{Z}\\}\) d) \(R=\\{a+b \sqrt{2}+c \sqrt{3} \mid a \in \mathbf{Z}, b, c \in \mathbf{Q}\\}\)

Let \(R=\left\\{a+b i \mid a, b \in \mathbf{Z}, i^{2}=-1\right\\}\), with addition and multiplication defined by \((a+b i)+(c+d i)=(a+c)+\) \((b+d) i\) and \((a+b i)(c+d i)=(a c-b d)+(b c+a d) i\), respectively. (a) Verify that \(R\) is an integral domain. (b) Determine all units in \(R\).

Write a computer program (or develop an algorithm) that reverses the order of the digits in a given positive integer. For example, the input 1374 should result in the output 4731 .

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

a) In how many ways can one select two positive integers \(m, n\), not necessarily distinct, so that \(1 \leq m \leq 100\), \(1 \leq n \leq 100\) and the last digit of \(7^{m}+3^{n}\) is \(8 ?\) b) Answer part (a) for the case where \(1 \leq m \leq 125,1 \leq\) \(n \leq 125 .\) c) If one randomly selects \(m, n\) [as in part (a)], what is the probability that 2 is now the last digit of \(7^{m}+3^{n}\) ?
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