Let R=ℤ_12 be the ring of integers modulo 12. If I=(2) is an ideal in ℤ_12 generated by the element 2, then to which ring is R/I isomorphic to? a) ℤ_2 b) ℤ_3 c) ℤ_4 d) ℤ_6 e) none of the above 2. Which is not a prime ideal in ℤ
Let R=ℤ_12 be the ring of integers modulo 12. If I=(2) is an ideal in ℤ_12 generated by the element 2, then to which ring is R/I isomorphic to? a) ℤ_2 b) ℤ_3 c) ℤ_4 d) ℤ_6 e) none of the above 2. Which is not a prime ideal in ℤ
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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Question
1. Let R=ℤ_12 be the ring of integers modulo 12. If I=(2) is an ideal in ℤ_12 generated by the element 2, then to which ring is R/I isomorphic to?
a) ℤ_2
b) ℤ_3
c) ℤ_4
d) ℤ_6
e) none of the above
2. Which is not a prime ideal in ℤ?
2. Which is not a prime ideal in ℤ?
a) {0}
b) ℤ
c) 2ℤ
d) 3ℤ
e) none of the above
3. Let R be a ring such that 3r=0 for all nonzero r but 2r≠0. Then which is definitely true?
3. Let R be a ring such that 3r=0 for all nonzero r but 2r≠0. Then which is definitely true?
a) R is a finite integral domain.
b) R only has no nontrivial ideals.
c) There is a subring of R which is isomorphic to ℤ_3.
d) all of the above
e) none of the above
4. Which among the following is true about the mapping f: ℤ_4 → ℤ_6 defined by f(x)=0 for all x ∈ ℤ_4.
4. Which among the following is true about the mapping f: ℤ_4 → ℤ_6 defined by f(x)=0 for all x ∈ ℤ_4.
a) f is a monomorphism
b) f is a epimorphism
c) f is a isomorphism
d) f is a endomorphism
e) none of the above
5. Which among the following mapping is a homomorphism?
5. Which among the following mapping is a homomorphism?
Remark: f:ℤ_n → ℤ_m means we are mapping integers modulo n to integers modulo m.
If it is not a homomorphism either addition or multiplication is not preserved.
So, either f(n+m)≠f(n)+f(m) or f(nm)≠f(n)f(m) for some n,m in ℤ_n
If it is not a homomorphism either addition or multiplication is not preserved.
So, either f(n+m)≠f(n)+f(m) or f(nm)≠f(n)f(m) for some n,m in ℤ_n
a) f: ℤ_5 → ℤ_10 where f(x)=5x
b) f: ℤ_4 → ℤ_12 where f(x)=3x
c) f: ℤ_12→ ℤ_30 where f(x)=10x
d) all of the above
e) none of the above
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