8.38. Find the characteristic of each of the following rings. 1. Z4 O Z10 2. M2(Z3)
8.38. Find the characteristic of each of the following rings. 1. Z4 O Z10 2. M2(Z3)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
100%
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15ed467-90ec-4e60-afef-3d3f6119f74d%2Fe678b59a-9112-4303-8582-75c6c6fd3da6%2Fpt9ts9_processed.png&w=3840&q=75)
Transcribed Image Text:**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.
Transcribed Image Text:**Exercise 8.38:**
Find the characteristic of each of the following rings.
1. \( \mathbb{Z}_4 \oplus \mathbb{Z}_{10} \)
2. \( M_2(\mathbb{Z}_3) \)
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 1 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
