please solve question 1
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter6: More On Rings
Section6.4: Maximal Ideals (optional)
Problem 5E
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please solve question 1
![1) Let Z be a ring; Then one of the following is False
:
a) The only idempotents in Z are {0, 1}
b) All non-zero elements of Z are non-zero divisors.
c) Z is an ideal in Q
d) Z is an Integral Domain
2) Given Gaussian Integers Z[i] and Z[√√-3] Then One of the following is False :
a) All numbers 5 & 7& 13 are irreducible elements in Z[√√-3]
b) Both 7 & 13 are reducible elements in Z[√-3] and 5 is reducible in Z[i]
c) The number 5 is an irreducible element Z[√√-3] but 5 is reducible in Z[i].
d) Both 5 &13 are reducible elements in Z[i].
3) One of the following is True:
a) Every infinite integral Domain is a field.
b) b)[13-2
is a non-zero divisor in M2x2
c) There is c0 in Zs such that Zs[x]/ x 2 +3 x +2c is a field.
d)x²-3x-4 =0 has no solution in Z7
4) Let I) 2x-10 in Z[x] II) 2x-10in [x] III) 2x-10 in Z₁₂[x] One of the following is TRUE :
a) I & II & III are irreducibles
b) Only I & II are irreducibles
c) Only I & III are irreducibles
d) Only II is irreducible
5) All non-zero divisors in Z[i] are
a) {1, -1} ONLY b){1,-1,i,-i}ONLY
c) {i, -i} ONLY d) All non-zero elements in Z[i].
6) One of the following is principal ideal but not prime ideal in Z:
a) <29>
b) <13>
c) <0>
d) <21>
7) Given
a) {1,-1}
:Z[i]Z where
b) {0}
(a+bi)=a² + b² . Then the kernel =
c) {1, -1, i, -i} d) {i, -i}
8) ) Let A=[0 ], B=[]=[!]
Then one of the following is TRUE
a) A &B& C are nilpotent in M₂(R)
b) A &B are nilpotent in M₂(R) but not C.
c) A & C are nilpotent in M₂(R) but not B
d) B& C are nilpotent in M₂(R) but not A.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe8d2d027-204d-466e-850d-579ecb98c79e%2Fed5181c3-3c97-4103-ae97-de9d959e41aa%2F2lk7rkn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1) Let Z be a ring; Then one of the following is False
:
a) The only idempotents in Z are {0, 1}
b) All non-zero elements of Z are non-zero divisors.
c) Z is an ideal in Q
d) Z is an Integral Domain
2) Given Gaussian Integers Z[i] and Z[√√-3] Then One of the following is False :
a) All numbers 5 & 7& 13 are irreducible elements in Z[√√-3]
b) Both 7 & 13 are reducible elements in Z[√-3] and 5 is reducible in Z[i]
c) The number 5 is an irreducible element Z[√√-3] but 5 is reducible in Z[i].
d) Both 5 &13 are reducible elements in Z[i].
3) One of the following is True:
a) Every infinite integral Domain is a field.
b) b)[13-2
is a non-zero divisor in M2x2
c) There is c0 in Zs such that Zs[x]/ x 2 +3 x +2c is a field.
d)x²-3x-4 =0 has no solution in Z7
4) Let I) 2x-10 in Z[x] II) 2x-10in [x] III) 2x-10 in Z₁₂[x] One of the following is TRUE :
a) I & II & III are irreducibles
b) Only I & II are irreducibles
c) Only I & III are irreducibles
d) Only II is irreducible
5) All non-zero divisors in Z[i] are
a) {1, -1} ONLY b){1,-1,i,-i}ONLY
c) {i, -i} ONLY d) All non-zero elements in Z[i].
6) One of the following is principal ideal but not prime ideal in Z:
a) <29>
b) <13>
c) <0>
d) <21>
7) Given
a) {1,-1}
:Z[i]Z where
b) {0}
(a+bi)=a² + b² . Then the kernel =
c) {1, -1, i, -i} d) {i, -i}
8) ) Let A=[0 ], B=[]=[!]
Then one of the following is TRUE
a) A &B& C are nilpotent in M₂(R)
b) A &B are nilpotent in M₂(R) but not C.
c) A & C are nilpotent in M₂(R) but not B
d) B& C are nilpotent in M₂(R) but not A.
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