Could you explain how to show 8.32 in detail? I also attached definitions and theorems in my textbook. **8.28.** For each of the following rings, which elements are units? Which are zero divisors? 1. \( \mathbb{Z}_{18} \) 2. \( \mathbb{Z}_3 \oplus \mathbb{Z}_9 \)**8.32.** Let \( R \) and \( S \) be rings. Under precisely what circumstances is \( R \oplus S \) an integral domain?**8.33.** Let \( F \) be a field with subfields \( K \) and \( L \). Show that \( K \cap L \) is a subfield of \( F \). Extend this to show that the intersection of any collection of subfields is a subfield.**8.34.** Let \( p \) be a prime and \( F \) a field with \( p^2 \) elements. Show that \( F \) cannot have more than one proper subfield. **Definition 8.8.** Let \( R \) be a commutative ring. Then a nonzero element \( a \in R \) is said to be a **zero divisor** if there exists a nonzero \( b \in R \) such that \( ab = 0 \).**Example 8.19.** In \( \mathbb{Z}_6 \), we note that 4 is a zero divisor, as \( 4 \cdot 3 = 0 \). On the other hand, 5 is not a zero divisor.**Example 8.20.** The ring of integers has no zero divisors.**Definition 8.9.** An **integral domain** is a commutative ring \( R \) with identity \( 1 \neq 0 \) having no zero divisors.**Example 8.21.** The rings \( \mathbb{Z}, \mathbb{Q}, \mathbb{R} \) and \( \mathbb{C} \) are all integral domains.**Example 8.22.** The polynomial ring \( \mathbb{R}[x] \) is an integral domain. Indeed, we know that it is a commutative ring with identity. Also, if \( f(x) = a_0 + a_1x + \cdots + a_nx^n \) and \( g(x) = b_0 + b_1x + \cdots + b_mx^m \), with \( a_i, b_i \in R \) and \( a_n \neq 0 \neq b_m \), then the unique term of highest degree in \( f(x)g(x) \) is \( a_nb_mx^{m+n} \). As \( R \) is an integral domain, \( a_nb_m \neq 0 \). Thus, \( f(x)g(x) \) is not the zero polynomial.**Example 8.23.** The rings \( \mathbb{Z}\mathbb{Z}, \mathbb{Z}_6 \) and \( M_2(\mathbb{R}) \) all fail to be integral domains. The first lacks an identity, the second has zero divisors and the third is not commutative.**Theorem 8.7 (Cancellation Law).** Let \( R \

Let R and S be rings. Find the conditions for R⊕S to be an integral domain.

Answered: 8.32. Let R and S be rings. Under…

8.32. Let R and S be rings. Under precisely what circumstances is ROS an integral domain?

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Thank you, I understood clearly and followed everything. However my textbook answer says 207pi. Could the textbook be wrong?