I is an ideal of Z[x].
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Could you explain how to show 9.2 in detail? I included list of theorems and definitions from the textbook.
![**Definition 9.1.** Let \( R \) be a ring. Then a subring \( I \) of \( R \) is said to be an ideal if \( ir, ri \in I \) for all \( i \in I \) and \( r \in R \). We call this the absorption property.
**Theorem 9.1.** Let \( R \) be a ring and \( I \) a subset of \( R \). Then \( I \) is an ideal if and only if
1. \( 0 \in I; \)
2. \( i - j \in I \) for all \( i, j \in I; \) and
3. \( ir, ri \in I \) for all \( i \in I, r \in R. \)
**Example 9.1.** Let \( n \) be any integer. Then \( n\mathbb{Z} \) is an ideal of \( \mathbb{Z} \). Indeed, we already know that it is a subring. But also, if \( nk \in n\mathbb{Z} \), then for any integer \( r, r(nk) = n(rk) \in n\mathbb{Z}. \)
**Example 9.2.** Let \( I \) be the set of all polynomials \( f(x) \in \mathbb{R}[x] \) such that \( f(0) = 0. \) We claim that \( I \) is an ideal in \( \mathbb{R}[x] \). Certainly, \( I \) contains the zero polynomial. Also, if \( f(0) = g(0) = 0 \), then \( (f-g)(0) = f(0) - g(0) = 0 \), hence \( f(x) - g(x) \in I. \) Also, if \( f(0) = 0 \) and \( h(x) \in \mathbb{R}[x], \) then \( h(0)f(0) = h(0)0 = 0. \) Hence, \( h(x)f(x) \in I. \)
**Example 9.3.** Let \( I \) be the set of all polynomials in \( \math](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15ed467-90ec-4e60-afef-3d3f6119f74d%2Fb65a524a-521c-4dc8-8f1f-6dbe441a4693%2F330sxp2_processed.png&w=3840&q=75)
Transcribed Image Text:**Definition 9.1.** Let \( R \) be a ring. Then a subring \( I \) of \( R \) is said to be an ideal if \( ir, ri \in I \) for all \( i \in I \) and \( r \in R \). We call this the absorption property.
**Theorem 9.1.** Let \( R \) be a ring and \( I \) a subset of \( R \). Then \( I \) is an ideal if and only if
1. \( 0 \in I; \)
2. \( i - j \in I \) for all \( i, j \in I; \) and
3. \( ir, ri \in I \) for all \( i \in I, r \in R. \)
**Example 9.1.** Let \( n \) be any integer. Then \( n\mathbb{Z} \) is an ideal of \( \mathbb{Z} \). Indeed, we already know that it is a subring. But also, if \( nk \in n\mathbb{Z} \), then for any integer \( r, r(nk) = n(rk) \in n\mathbb{Z}. \)
**Example 9.2.** Let \( I \) be the set of all polynomials \( f(x) \in \mathbb{R}[x] \) such that \( f(0) = 0. \) We claim that \( I \) is an ideal in \( \mathbb{R}[x] \). Certainly, \( I \) contains the zero polynomial. Also, if \( f(0) = g(0) = 0 \), then \( (f-g)(0) = f(0) - g(0) = 0 \), hence \( f(x) - g(x) \in I. \) Also, if \( f(0) = 0 \) and \( h(x) \in \mathbb{R}[x], \) then \( h(0)f(0) = h(0)0 = 0. \) Hence, \( h(x)f(x) \in I. \)
**Example 9.3.** Let \( I \) be the set of all polynomials in \( \math
![**9.2.** Let \( I \) be the set of all polynomials in \(\mathbb{Z}[x]\) whose constant term is a multiple of 5. Show that \( I \) is an ideal of \(\mathbb{Z}[x]\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15ed467-90ec-4e60-afef-3d3f6119f74d%2Fb65a524a-521c-4dc8-8f1f-6dbe441a4693%2Fwpavwoq_processed.png&w=3840&q=75)
Transcribed Image Text:**9.2.** Let \( I \) be the set of all polynomials in \(\mathbb{Z}[x]\) whose constant term is a multiple of 5. Show that \( I \) is an ideal of \(\mathbb{Z}[x]\).
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