Each of the quotient rings R/I on the left is isomorphic to one of the rings S on the right. Match each ring with its isomorphic partner, and prove that they really are isomorphic by constructing a surjective ring homomorphism p: RS with kernel I. R/I S A) Z[X]/(8, 12, X) 1. ZZ 2. Q 3. R B) Q[X]/(X² - 2) 4. C 5. F4 C) R[X]/(X-√2) 6. F8 7. ZA 8. Zg D) R[X]/(x²+X+2) 9. {a+b√2 a,bQ} 10. a {( 8 b ) | ab € F₁ } a E) R[X]/(X²) F) Z₁₂[X]/(X + X²+1) a 11. | a, b € F8 0 a 12. {( o b) | ab € R } a 0 a In examples 5, 6, 10, and 11 the symbols F4 and Fg denote a field of 4 and a field of 8 elements respectively. In examples 10, 11, and 12 the ring operations are addition and multiplication of matrices. Some methods you might use to try and match up R/I and S: (i) guess, (ii) think about representatives in the quotient ring, and what the multiplication rules are and try and match that up with something in the S column, (ii) try and make up a homomorphism from R to somewhere that would have elements of the ideal I in the kernel.
Each of the quotient rings R/I on the left is isomorphic to one of the rings S on the right. Match each ring with its isomorphic partner, and prove that they really are isomorphic by constructing a surjective ring homomorphism p: RS with kernel I. R/I S A) Z[X]/(8, 12, X) 1. ZZ 2. Q 3. R B) Q[X]/(X² - 2) 4. C 5. F4 C) R[X]/(X-√2) 6. F8 7. ZA 8. Zg D) R[X]/(x²+X+2) 9. {a+b√2 a,bQ} 10. a {( 8 b ) | ab € F₁ } a E) R[X]/(X²) F) Z₁₂[X]/(X + X²+1) a 11. | a, b € F8 0 a 12. {( o b) | ab € R } a 0 a In examples 5, 6, 10, and 11 the symbols F4 and Fg denote a field of 4 and a field of 8 elements respectively. In examples 10, 11, and 12 the ring operations are addition and multiplication of matrices. Some methods you might use to try and match up R/I and S: (i) guess, (ii) think about representatives in the quotient ring, and what the multiplication rules are and try and match that up with something in the S column, (ii) try and make up a homomorphism from R to somewhere that would have elements of the ideal I in the kernel.
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
![Each of the quotient rings R/I on the left is isomorphic to one of the rings S on the right.
Match each ring with its isomorphic partner, and prove that they really are isomorphic by
constructing a surjective ring homomorphism p: RS with kernel I.
R/I
S
A) Z[X]/(8, 12, X)
1. ZZ
2. Q
3. R
B) Q[X]/(X² - 2)
4. C
5. F4
C) R[X]/(X-√2)
6. F8
7. ZA
8. Zg
D) R[X]/(x²+X+2)
9. {a+b√2 a,bQ}
10.
a
{( 8 b ) | ab € F₁ }
a
E) R[X]/(X²)
F) Z₁₂[X]/(X + X²+1)
a
11.
| a, b € F8
0 a
12.
{( o b) | ab € R }
a
0 a
In examples 5, 6, 10, and 11 the symbols F4 and Fg denote a field of 4 and a field of 8
elements respectively.
In examples 10, 11, and 12 the ring operations are addition and multiplication of matrices.
Some methods you might use to try and match up R/I and S: (i) guess, (ii) think about
representatives in the quotient ring, and what the multiplication rules are and try and match
that up with something in the S column, (ii) try and make up a homomorphism from R to
somewhere that would have elements of the ideal I in the kernel.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf58e778-209d-4051-b758-0c57e3f8187c%2Fd70019eb-53b2-486d-bcf6-2808edb37aaf%2F18g6yku_processed.png&w=3840&q=75)
Transcribed Image Text:Each of the quotient rings R/I on the left is isomorphic to one of the rings S on the right.
Match each ring with its isomorphic partner, and prove that they really are isomorphic by
constructing a surjective ring homomorphism p: RS with kernel I.
R/I
S
A) Z[X]/(8, 12, X)
1. ZZ
2. Q
3. R
B) Q[X]/(X² - 2)
4. C
5. F4
C) R[X]/(X-√2)
6. F8
7. ZA
8. Zg
D) R[X]/(x²+X+2)
9. {a+b√2 a,bQ}
10.
a
{( 8 b ) | ab € F₁ }
a
E) R[X]/(X²)
F) Z₁₂[X]/(X + X²+1)
a
11.
| a, b € F8
0 a
12.
{( o b) | ab € R }
a
0 a
In examples 5, 6, 10, and 11 the symbols F4 and Fg denote a field of 4 and a field of 8
elements respectively.
In examples 10, 11, and 12 the ring operations are addition and multiplication of matrices.
Some methods you might use to try and match up R/I and S: (i) guess, (ii) think about
representatives in the quotient ring, and what the multiplication rules are and try and match
that up with something in the S column, (ii) try and make up a homomorphism from R to
somewhere that would have elements of the ideal I in the kernel.
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