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Chapter 14: Problem 4

If \(S=\left\\{\left[\begin{array}{ll}a & 0 \\ 0 & a\end{array}\right] \mid a \in \mathbf{R}\right\\}\), then \(S\) is a ring under matrix addition and multiplication. Prove that \(\mathbf{R}\) is isomorphic to \(S\).

Short Answer

Expert verified
The proposed function \(f: \mathbf{R} \rightarrow S\), where \(f(a) = \left[\begin{array}{ll}a & 0 \ 0 & a\end{array}\right]\) is a ring isomorphism. It is a bijection and it preserves both the addition and the multiplication operations of the rings, thus proving that \( \mathbf{R} \) is isomorphic to \( S \).

Step by step solution

01

Define the terms

We start by defining what it means for two rings to be isomorphic. Two rings A and B are isomorphic if there is a bijective function \( f: A \rightarrow B \) such that for any \( a, b \in A \), \( f(a+b) = f(a) + f(b) \) and \( f(ab) = f(a)f(b) \) (preservation of addition and multiplication). In this problem, our rings are \( \mathbf{R} \) and \( S \).
02

Define the function \(f\)

Let's define a function \(f: \mathbf{R} \rightarrow S\) by \( f(a) = \left[\begin{array}{ll}a & 0 \ 0 & a\end{array}\right] \). We need to prove that this function is a bijection and that it preserves addition and multiplication.
03

Verify that \(f\) is a bijection

To show that \(f\) is a bijection, we need to prove that it is both injective (one-to-one) and surjective (onto). It is clear that \(f\) is injective: if \(f(a) = f(b)\) for \(a, b \in \mathbf{R}\), then \(a = b\). Also, \(f\) is surjective: for each \(\left[\begin{array}{ll}a & 0 \ 0 & a\end{array}\right] \in S\), we can choose \(a \in \mathbf{R}\) such that \(f(a) = \left[\begin{array}{ll}a & 0 \ 0 & a\end{array}\right]\), so \(f\) is a bijection.
04

Verify that \(f\) preserves addition

For any \(a, b \in \mathbf{R}\), we need to check that \(f(a + b) = f(a) + f(b)\). Indeed, \(f(a + b) = \left[\begin{array}{ll}a+b & 0 \ 0 & a+b\end{array}\right]\) and \(f(a) + f(b) = \left[\begin{array}{ll}a & 0 \ 0 & a\end{array}\right] + \left[\begin{array}{ll}b & 0 \ 0 & b\end{array}\right] = \left[\begin{array}{ll}a+b & 0 \ 0 & a+b\end{array}\right]\), so \(f\) preserves addition.
05

Verify that \(f\) preserves multiplication

For any \(a, b \in \mathbf{R}\), we need to check that \(f(ab) = f(a)f(b)\). Indeed, \(f(ab) = \left[\begin{array}{ll}ab & 0 \ 0 & ab\end{array}\right]\) and \(f(a)f(b) = \left[\begin{array}{ll}a & 0 \ 0 & a\end{array}\right]\left[\begin{array}{ll}b & 0 \ 0 & b\end{array}\right] = \left[\begin{array}{ll}ab & 0 \ 0 & ab\end{array}\right]\), so \(f\) preserves multiplication.

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Most popular questions from this chapter

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

Let \(k, m\) be fixed integers. Find all values for \(k, m\) for which \((\mathbf{Z}, \oplus, \odot)\) is a ring under the binary operations \(x \oplus y=\) \(x+y-k, x \odot y=x+y-m x y\), where \(x, y \in \mathbf{Z}\).

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a) How many units are there in \(\mathbf{Z}_{15}\) ? How many in \(\mathbf{Z}_{3} \times \mathbf{Z}_{5} ?\) b) Are \(\mathbf{Z}_{15}\) and \(\mathbf{Z}_{3} \times \mathbf{Z}_{5}\) isomorphic?

Given a finite field \(F\), let \(M_{2}(F)\) denote the set of all \(2 \times 2\) matrices with entries from \(F\). As in Example 14.2, \(\left(M_{2}(F),+, \cdot\right)\) becomes a noncommutative ring with unity. a) Determine the number of elements in \(M_{2}(F)\) if \(F\) is i) \(\mathbf{Z}_{2}\) ii) \(\mathbf{Z}_{3}\) iii) \(\mathbf{Z}_{p}, p\) a prime b) As in Exercise 13 of Section 14.1, \(A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] \in\) \(M_{2}\left(\mathbf{Z}_{p}\right)\) is a unit if and only if \(a d-b c \neq z\). This occurs if the first row of \(A\) does not contain all zeros (that is, \(z\) 's) and the second row is not a multiple (by an element of \(\mathbf{Z}_{p}\) ) of the first. Use this observation to determine the number of units in i) \(M_{2}\left(\mathbf{Z}_{2}\right)\) ii) \(M_{2}\left(\mathbf{Z}_{3}\right)\) iii) \(M_{2}\left(\mathbf{Z}_{p}\right), p\) a prime
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