Could you explain how to show 8.38 in detail? I also attached definitions and theorems in my textbook. **Definition 8.13.** Let \( R \) be a ring. Then the **characteristic** of \( R \), denoted char \( R \), is the smallest positive integer \( n \) such that \( na = 0 \) for all \( a \in R \). If no such \( n \) exists, then char \( R = 0 \).**Example 8.31.** The characteristic of \( \mathbb{Z}_n \) is \( n \), as clearly \( na = 0 \) for any \( a \in \mathbb{Z}_n \), whereas no smaller value than \( n \) will work if we take \( a = 1 \).**Example 8.32.** The ring of integers has characteristic zero.In fact, for rings with identity, we only need to look at the identity.**Theorem 8.13.** Let \( R \) be a ring with identity. Regarding \( R \) as an additive group, if the order of 1 is \( n < \infty \), then \( R \) has characteristic \( n \). If 1 has infinite order, then \( R \) has characteristic zero.**Corollary 8.2.** Let \( R \) be a ring with identity. Then every unital subring of \( R \) has the same characteristic as \( R \).**Theorem 8.14 (Freshman’s Dream).** Let \( R \) be a commutative ring of prime characteristic \( p \). Then for any \( a, b \in R \), we have\[(a + b)^p = a^p + b^p.\]**Theorem 8.15.** The characteristic of an integral domain is either zero or a prime. **Exercise 8.38:**Find the characteristic of each of the following rings.1. \( \mathbb{Z}_4 \oplus \mathbb{Z}_{10} \)2. \( M_2(\mathbb{Z}_3) \)

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Answered: 8.38. Find the characteristic of each…

8.38. Find the characteristic of each of the following rings. 1. Z4 O Z10 2. M2(Z3)

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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