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].

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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please solve question 2 

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.
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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