(b) (5 Pts) Prove that o is surjective onto its image. Answer
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= a (mod n) is a ring homomorphism. (10) Suppose that o Z Z defined by...
Please solve all questions 1. Let 0 : Z/9Z+Z/12Z be the map 6(x + 9Z) =...
Please solve all questions
1. Let 0 : Z/9Z+Z/12Z be the map 6(x + 9Z) = 4.+ 12Z (a) Prove that o is a ring homomorphism. Note: You must first show that o is well-defined (b) Is o injective? explain (c) Is o surjective? explain 2. In Z, let I = (3) and J = (18). Show that the group I/J is isomorphic to the group Z6 but that the ring I/J is not ring-isomorphic to the ring Z6. 3....
(3.) (a) Suppose that y: R S is a ring homomorphism. Please prove that (-a) =...
(3.) (a) Suppose that y: R S is a ring homomorphism. Please prove that (-a) = -f(a) for all a ER (b) Suppose R and S are rings. Define the zero function y: R S by pa) = Os for all GER. Is y a ring homomorphism? Please explain. (4.) Suppose that p is a prime number and 4: Z, Z, is defined by wa) = a.
a. Suppose that 9: Z Z , is defined by p(x)= x mod 4. ind the...
a. Suppose that 9: Z Z , is defined by p(x)= x mod 4. ind the kernel of this mapping and explain why this mapping is NOT a homomorphism. nt. What do we know about the kernel of a homomorphism.
Question 5 (6 points) Let o: R+S be a ring homomorphism. Suppose that o(R) and ker...
Question 5 (6 points) Let o: R+S be a ring homomorphism. Suppose that o(R) and ker o contain no nonzero nilpotent elements. Prove that contains no nonzero nilpotent elements.
9. Suppose n : (Z4 ⓇZ3) + (Z2 Z3) is defined by n(a,b) = (a mod...
9. Suppose n : (Z4 ⓇZ3) + (Z2 Z3) is defined by n(a,b) = (a mod 2, 0). This is a homomorphism. (You can believe me about that, and you don't need to check.) Write down the elements of the kernel of n.
(4) Let p Z be a prime. Prove that zli/(p+1) has exactly ] p2 +1 elements. Use that 5+5i (2+i)(3+...
(4) Let p Z be a prime. Prove that zli/(p+1) has exactly ] p2 +1 elements. Use that 5+5i (2+i)(3+i) to determine how many elements Zu/5+5i) has. (5) Let m,n be integers with m|n. Prove that the surjective ring homomor- phism Z/n -> Z/m induces a group homomorphism on units, and that this group homomorphism is also surjective.
(4) Let p Z be a prime. Prove that zli/(p+1) has exactly ] p2 +1 elements. Use that 5+5i (2+i)(3+i) to determine...
18. Let o: R+ S be a ring homomorphism. Prove each of the following statements. (a)...
18. Let o: R+ S be a ring homomorphism. Prove each of the following statements. (a) If R is a commutative ring, then (R) is a commutative ring. (b) (0)=0. (c) Let 18 and 1s be the identities for R and S, respectively. If o is onto, then (1r) = 1s. (d) If R is a field and $(R) +0, then (R) is a field.
1. a) Let A = {2n|n ∈ ℤ} (ie, A is the set of even numbers)...
1. a) Let A = {2n|n ∈ ℤ} (ie, A is the set of even numbers) and define function f: ℝ → {0,1}, where f(x) = XA(x) That is, f is the characteristic function of set A; it maps elements of the domain that are in set A (ie, those that are even integers) to 1 and all other elements of the domain to 0. By demonstrating a counter-example, show that the function f is not injective (not one-to-one). b)...
1- (2,5+2,5 mark) Consider in GL(2, Q), the subset (a=1 or a=-1),bez Prove that H, with multiplication, is a subgroup of GL(2,Q) a) Is the function b) an homomorphism of groups? Justify your answ...
1- (2,5+2,5 mark) Consider in GL(2, Q), the subset (a=1 or a=-1),bez Prove that H, with multiplication, is a subgroup of GL(2,Q) a) Is the function b) an homomorphism of groups? Justify your answer 2 (3 marks) Let G be a group and a E G an element of order 12. Find the orders of each of the elements of (a) 3- (1+1,5 marks) Let G be a group such that any non-identity element has order 2. Prove that a)...
quention for 8 iz) 23)1Dy ave 7. (10M) Prove that o: Z x Z Z given...
quention for 8
iz) 23)1Dy ave 7. (10M) Prove that o: Z x Z Z given by (a, b) a+b homomorphism and find its kernel. Describe the set is a 8. (10M) Prove that there is no homomorphism from Zs x Z2 onto Z4 x Z 9.(10M) Let G be a order of the element gH in G/H must divide the order of g in G. finite group and let H be a normal subgroup of G. Prove that (16M)...
Please solve all questions
1. Let 0 : Z/9Z+Z/12Z be the map 6(x + 9Z) = 4.+ 12Z (a) Prove that o is a ring homomorphism. Note: You must first show that o is well-defined (b) Is o injective? explain (c) Is o surjective? explain 2. In Z, let I = (3) and J = (18). Show that the group I/J is isomorphic to the group Z6 but that the ring I/J is not ring-isomorphic to the ring Z6. 3....
(3.) (a) Suppose that y: R S is a ring homomorphism. Please prove that (-a) = -f(a) for all a ER (b) Suppose R and S are rings. Define the zero function y: R S by pa) = Os for all GER. Is y a ring homomorphism? Please explain. (4.) Suppose that p is a prime number and 4: Z, Z, is defined by wa) = a.
a. Suppose that 9: Z Z , is defined by p(x)= x mod 4. ind the kernel of this mapping and explain why this mapping is NOT a homomorphism. nt. What do we know about the kernel of a homomorphism.
Question 5 (6 points) Let o: R+S be a ring homomorphism. Suppose that o(R) and ker o contain no nonzero nilpotent elements. Prove that contains no nonzero nilpotent elements.
9. Suppose n : (Z4 ⓇZ3) + (Z2 Z3) is defined by n(a,b) = (a mod 2, 0). This is a homomorphism. (You can believe me about that, and you don't need to check.) Write down the elements of the kernel of n.
(4) Let p Z be a prime. Prove that zli/(p+1) has exactly ] p2 +1 elements. Use that 5+5i (2+i)(3+i) to determine how many elements Zu/5+5i) has. (5) Let m,n be integers with m|n. Prove that the surjective ring homomor- phism Z/n -> Z/m induces a group homomorphism on units, and that this group homomorphism is also surjective.
(4) Let p Z be a prime. Prove that zli/(p+1) has exactly ] p2 +1 elements. Use that 5+5i (2+i)(3+i) to determine...
18. Let o: R+ S be a ring homomorphism. Prove each of the following statements. (a) If R is a commutative ring, then (R) is a commutative ring. (b) (0)=0. (c) Let 18 and 1s be the identities for R and S, respectively. If o is onto, then (1r) = 1s. (d) If R is a field and $(R) +0, then (R) is a field.
1- (2,5+2,5 mark) Consider in GL(2, Q), the subset (a=1 or a=-1),bez Prove that H, with multiplication, is a subgroup of GL(2,Q) a) Is the function b) an homomorphism of groups? Justify your answer 2 (3 marks) Let G be a group and a E G an element of order 12. Find the orders of each of the elements of (a) 3- (1+1,5 marks) Let G be a group such that any non-identity element has order 2. Prove that a)...
quention for 8
iz) 23)1Dy ave 7. (10M) Prove that o: Z x Z Z given by (a, b) a+b homomorphism and find its kernel. Describe the set is a 8. (10M) Prove that there is no homomorphism from Zs x Z2 onto Z4 x Z 9.(10M) Let G be a order of the element gH in G/H must divide the order of g in G. finite group and let H be a normal subgroup of G. Prove that (16M)...
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