Let R be a ring and A a set, as above. (a) Prove the associativity of multiplication on Rª (b) Which elements of RA serve as the additive and multiplicative identity elements? Justify your answers.
Let R be a ring and A a set, as above. (a) Prove the associativity of multiplication on Rª (b) Which elements of RA serve as the additive and multiplicative identity elements? Justify your answers.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Problem 6 Please
![### Problem 6
Let \( R \) be a ring and \( A \) a set, as above.
(a) Prove the associativity of multiplication on \( R^A \).
(b) Which elements of \( R^A \) serve as the additive and multiplicative identity elements? Justify your answers.
### Suggested Solutions
#### (a) Prove the Associativity of Multiplication on \( R^A \)
To prove the associativity of multiplication on \( R^A \), consider three elements \( f, g, h \in R^A \). For each \( a \in A \):
\[
(fg)h(a) = (f(a)g(a))h(a)
\]
Since \( g(a) \) and \( h(a) \) are elements of a ring \( R \) and multiplication in \( R \) is associative, we have:
\[
f(a)(g(a)h(a)) = (f(a)g(a))h(a)
\]
Thus, we conclude:
\[
f(gh)(a) = f(a)(g(a)h(a))
\]
Given that \( a \in A \) was arbitrary, it follows that multiplication in \( R^A \) is associative:
\[
f(gh) = (fg)h \quad \text{for all} \quad f, g, h \in R^A
\]
#### (b) Identity Elements in \( R^A \)
**Additive Identity:**
The additive identity in \( R \) is typically denoted as 0. In \( R^A \), the function \( f: A \to R \) where \( f(a) = 0 \) for all \( a \in A \) serves as the additive identity. Let \( f \in R^A \):
\[
f + 0 = 0 + f = f
\]
Hence, the zero function \( 0_A \) in \( R^A \) is defined by:
\[
0_A(a) = 0 \quad \text{for all} \quad a \in A
\]
**Multiplicative Identity:**
The multiplicative identity in \( R \) is typically denoted as 1. In \( R^A \), the function \( 1_A: A \to R \), where \( 1_A(a) = 1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Febb913e1-4986-4d74-b6ce-ad576ddf43d3%2Fc949a23d-559d-462a-a7fb-51d4b2a83ce2%2F3v98j7_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 6
Let \( R \) be a ring and \( A \) a set, as above.
(a) Prove the associativity of multiplication on \( R^A \).
(b) Which elements of \( R^A \) serve as the additive and multiplicative identity elements? Justify your answers.
### Suggested Solutions
#### (a) Prove the Associativity of Multiplication on \( R^A \)
To prove the associativity of multiplication on \( R^A \), consider three elements \( f, g, h \in R^A \). For each \( a \in A \):
\[
(fg)h(a) = (f(a)g(a))h(a)
\]
Since \( g(a) \) and \( h(a) \) are elements of a ring \( R \) and multiplication in \( R \) is associative, we have:
\[
f(a)(g(a)h(a)) = (f(a)g(a))h(a)
\]
Thus, we conclude:
\[
f(gh)(a) = f(a)(g(a)h(a))
\]
Given that \( a \in A \) was arbitrary, it follows that multiplication in \( R^A \) is associative:
\[
f(gh) = (fg)h \quad \text{for all} \quad f, g, h \in R^A
\]
#### (b) Identity Elements in \( R^A \)
**Additive Identity:**
The additive identity in \( R \) is typically denoted as 0. In \( R^A \), the function \( f: A \to R \) where \( f(a) = 0 \) for all \( a \in A \) serves as the additive identity. Let \( f \in R^A \):
\[
f + 0 = 0 + f = f
\]
Hence, the zero function \( 0_A \) in \( R^A \) is defined by:
\[
0_A(a) = 0 \quad \text{for all} \quad a \in A
\]
**Multiplicative Identity:**
The multiplicative identity in \( R \) is typically denoted as 1. In \( R^A \), the function \( 1_A: A \to R \), where \( 1_A(a) = 1
![### Advanced Algebra: Binary Operations and Rings
#### Problem Statement
Let \( A \) be a set and let \( \beta \) be a binary operation on \( A \). We say that a subset \( B \subset A \) is **closed under \( \beta \)** if for all \( a_1, a_2 \in B \), \( a_1 \beta a_2 \in B \).
1. **Prove** that if subsets \( B \) and \( C \) of \( A \) are closed under \( \beta \), then the intersection, \( B \cap C \), is also closed under \( \beta \).
2. For \( a \in \mathbb{Z} \), define the set \( a\mathbb{Z} = \{ak : k \in \mathbb{Z}\} \). Prove that \( a\mathbb{Z} \) is closed under the addition operation on \( \mathbb{Z} \). (You can take for granted that the sum and product of two integers is an integer.)
#### Ring Structures
Let \( R \) be a ring and \( A \) a set. We denote by \( R^A \) the set of all functions from \( A \) to \( R \). We can use the ring structure on \( R \) to endow \( R^A \) with its own fabulous ring structure via pointwise addition and multiplication: for \( f, g \in R^A \), we define \( f+g : A \rightarrow R \) and \( fg : A \rightarrow R \) by the formulas:
\[ (f + g)(a) = f(a) + g(a) \]
and
\[ (fg)(a) = f(a)g(a) \]
for \( a \in A \). (Note that, in each of the displayed formulas, the arithmetic taking place on the right-hand side is that of the ring \( R \).)
#### Proposition
**Proposition:** *With the operations of addition and multiplication just defined, \( R^A \) is a ring.*
The proof of the proposition entails verifying properties (R1)–(R6). We will prove associativity of addition on \( R^A \) (half of (R1)), and you will prove associativity of multiplication (the other half of (R1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Febb913e1-4986-4d74-b6ce-ad576ddf43d3%2Fc949a23d-559d-462a-a7fb-51d4b2a83ce2%2Fagxzpxq_processed.png&w=3840&q=75)
Transcribed Image Text:### Advanced Algebra: Binary Operations and Rings
#### Problem Statement
Let \( A \) be a set and let \( \beta \) be a binary operation on \( A \). We say that a subset \( B \subset A \) is **closed under \( \beta \)** if for all \( a_1, a_2 \in B \), \( a_1 \beta a_2 \in B \).
1. **Prove** that if subsets \( B \) and \( C \) of \( A \) are closed under \( \beta \), then the intersection, \( B \cap C \), is also closed under \( \beta \).
2. For \( a \in \mathbb{Z} \), define the set \( a\mathbb{Z} = \{ak : k \in \mathbb{Z}\} \). Prove that \( a\mathbb{Z} \) is closed under the addition operation on \( \mathbb{Z} \). (You can take for granted that the sum and product of two integers is an integer.)
#### Ring Structures
Let \( R \) be a ring and \( A \) a set. We denote by \( R^A \) the set of all functions from \( A \) to \( R \). We can use the ring structure on \( R \) to endow \( R^A \) with its own fabulous ring structure via pointwise addition and multiplication: for \( f, g \in R^A \), we define \( f+g : A \rightarrow R \) and \( fg : A \rightarrow R \) by the formulas:
\[ (f + g)(a) = f(a) + g(a) \]
and
\[ (fg)(a) = f(a)g(a) \]
for \( a \in A \). (Note that, in each of the displayed formulas, the arithmetic taking place on the right-hand side is that of the ring \( R \).)
#### Proposition
**Proposition:** *With the operations of addition and multiplication just defined, \( R^A \) is a ring.*
The proof of the proposition entails verifying properties (R1)–(R6). We will prove associativity of addition on \( R^A \) (half of (R1)), and you will prove associativity of multiplication (the other half of (R1
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