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Chapter 14: Problem 32
Determine the last digit in \(3^{55}\).
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Define relation \(\mathscr{R}\) on \(\mathbf{Z}^{+}\)by \(a \mathscr{R} b\), if \(\tau(a)=\tau(b)\), where \(\tau(a)=\) the number of positive (integer) divisors of \(a\). For example, \(2 \mathscr{R} 3\) and \(4 \mathscr{R} 25\) but \(5 \not 9\). a) Verify that \(\mathscr{R}\) is an equivalence relation on \(\mathbf{Z}^{+}\). b) For the equivalence classes \([a]\) and \([b]\) induced by \(\mathscr{R}\), define operations of addition and multiplication by \([a]+\) \([b]=[a+b]\) and \([a][b]=[a b]\). Are these operations welldefined [that is, does \(a \mathscr{R} c, b \Re d \Rightarrow(a+b) \mathscr{R}(c+d)\), \((a b) \mathscr{R}(c d)] ?\)
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 \(S\).
a) For \(R=M_{2}(\mathbf{Z})\), prove that $$ S=\left\\{\left[\begin{array}{ll} a & \theta \\ 0 & 0 \end{array}\right] \mid a \in \mathbf{Z}\right\\} $$ is a subring of \(R\). b) What is the unity of \(R ?\) c) Does \(S\) have a unity? d) Does \(S\) have any properties that \(R\) does not have? e) Is \(S\) an ideal of \(R ?\)
Let \((R,+, \cdot)\) be a ring with \(a, b, c, d\) elements of \(R\). State the conditions (from the definition of a ring) that are needed to prove each of the following results. a) \((a+b)+c=b+(c+a)\) b) \(d+a(b+c)=a b+(d+a c)\) c) \(c(d+b)+a b=(a+c) b+c d\) d) \(a(b c)+(a b) d=(a b)(d+c)\)
Let \((R,+, \cdot)\) be a ring with unity \(u\), and \(|R|=8\). On \(R^{4}=R \times R \times R \times R\), define \(+\) and \(\cdot\) as suggested by Exercise 18. 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?
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