QUESTION 13 Prove or disprove that the map, f:Q →Q defined by f(x) = |x| for all xEQ , is a ring homomorphism.

Q: Prove or disprove that the map, f:0→0 defined by is a ring homomorphism. f(x) =|x| for all XEQ.

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Q: The function p: Zs → Z30 defined by ø(a) = 6a is a ring homomorphism. True False

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Q: a The mapping $ :C → M2 (R) given by (a + bi) = -b is a not one-to-one a ring homomorphism not a…

A: To find - The mapping ϕ : ℂ → M2ℝ given by ϕa + ib = ab-ba is ?     Definition used - A function is…

Q: 4) Let f : C – C be a homomorphism of rings with f(x) = x for all x E R. Show that f is either the…

A: If a function f is ring homomorphism from a ring R to another ring R'. Then We knew from the…

Q: Q₁: (A) Let R be a ring with identity. Define g: Z → Z by g(x) = x. 1, Vx € Z. Is g a homorphism?…

A: Ring homorphism

Q: a) Let Ø. R[x] R be a map given by Øf(x)) = f(1) Show that Ø is a ring homomorphism. b) Describe the…

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Q: If fis a ring homomorphism from Zm to Z, such that f(1)= b, then b**2 = b*. True False

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Q: 17. If E is an extension of F and f (x) e F[x] and if is an automorphism of E leaving every element…

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Q: Let R = Q[V2] and S = Q[V3]. Show that the only ring homomorphism from R to S is the trivial one. In…

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Q: If R is a ring and f(x) and g(x) are of degrees 3 and 4 then f(x)g(x) must have degree 7.

A: Given that R is a ring. deg{f(x)}=3 deg{g(x)}=4

Q: 1. Let R be a ring with the additive identity 0. Prove that for any a E R, 0- a = 0.

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Q: Which of the following maps C[X] → C[X] are ring homomorphisms? Select all that apply: O The map…

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Q: If E is an extension of F and f (x) e F[x] and if o is an automorphism of E leaving every element of…

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Q: - Let R be a commutative ring with unity of prime characteristic p. Show that the map op : R →R…

A: Homomorphic ring means F:R---->R' and a,  b belongs to R then f(a+b) = f(a)+(b)        and f(ab)…

Q: Question 4 Suppose there is a homomorphism y from a unital ring R to another ring S. Show that y(1)…

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Q: If is a homomorphism from the ring R to the ring R' , show that; a) (0)=0 b) (−r)= −(r)for all…

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Q: - The mapping f: Z → Z defined by f (n) = 5n is a ring homomorphism. a) True b) False

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Q: Q1: Let R be a commutative ring with Char(R) = 2 and let p:R → R be defined such that o (a) = a².…

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Q: If E is an extension of F and f(x) belong to F[x] and if phi is an automorphism of E leaving every…

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Q: = {I6 Tla, b e Q}and let p:R - Q be defined by : 0( 1) ois a ring a) Homomorphism. b) Isomorphism c)…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Let f:R, → R, and let g : R, → R, be two ring isomorphisms. If ø = g• f , then 2- (e2) =

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Q: Let f:R→s be a ring homomorphism. (i) Prove that if K is a subring of then f(K) is a subring of s.…

A: The solution is given as

Q: Let R be a ring. Consider the map Ø:Q[x]¬Q defined by ø(f(x))=f(3). Then, the Kernel of ø is: *…

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Q: If u is finitely additive on a ring R; E, F eR show p(E) +µ(F) = u(EJ F)+u(En F)

A: Here, we need to write the union of E and F as union of disjoint subsets then use the properties of…

Q: If fis a ring homomorphism from Zm to Z, such that f (1) = b, then b*+2 = b*. True False

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Q: Give an example (and justify your choice) of such a subring of a ring P[x] that contains P and is…

A: Given: Ring P[x] and a subring which contains P.

Q: 9. Show that the function f:x V-1 is a ring homomorphism f: Z[x] C. 10. What is the kernel of f?…

A: Given: The function, f:x↦-1, which is a ring homomorphism f:ℤx→ℂ.   To determine: The kernel of f…

Q: - The mapping f:Z → Z defined by f (n) = 5n is a ring homomorphism. a) True b) False

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Q: Let f : R S be a homomorphism of rings, 1. If K is a subringof R, Is o(K) a subring of S? If so,…

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Q: Let f:R→S be a ring homomorphism. (i) Prove that if K is a subring of R then fIK) is a subring of s-…

A: Suppose f:R→S be a ring homomorphism then ; fx1+x2=fx1+fx2 for all x1,x2∈R. fx1·x2=fx1·fx2 for all…

Q: Let R be a commutative ring such that 2x=0 xe R. Then the mapping f: R →→R defined as f(x) = x² is a…

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Q: 17. If E is an extension of F and f (x) e F [x] and if is an automorphism of E leaving every element…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 16. Let f: R S be a ring homomorphism with J an ideal of S. Define I= {r ER| f(r) € J} and prove…

A: Given :  f : R →S be a ring homomorphism with J an ideal of S. To prove :  I = r∈R : f(r)∈J  is an…

Q: Let f:R→s be a ring homomorphism. (1) Prove that if K is a subring of R then f(K) is a subring of s.…

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Q: 18. Let p:C + C be an isomorphism of rings such that e(a) - a for each ae Q. Suppose r E Cisa root…

A: Given: Let ϕ:ℂ→ℂ be an isomorphism or rings such that ϕ(a)=a for each a∈ℚ. Suppose r∈ℂ is the root…

Q: If f:(R.+) - (R', +') be a ring Homomorphism, and R is integral domain, then R' is integral domain…

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Q: QUESTION 10 Let f:R→s be a ring homomorphism. (i) Prove that if K is a subring of R then f(K) is a…

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Q: If u is finitely additive on a ring R; E, F eR show p(E) +u(F) = µ(B F)+µ(EnF)

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Q: QUESTION 7 1 Prove the o: R* → R* defined by o(x)=-is an isomorphism (R* is the nonzero real…

A: Solution

Q: The map f: Z→ Z,o given by f(x)= 2x is a ring homomorphism. Select one. True False

A: SINCE YOU HAVE ASKED MULTIPLE QUESTIONS IN SINGLE REQUEST, WE WILL BE ANSWERING ONLY THE FIRST…

Q: a Let S = { : a,b e R } . Show that Ø : C→ S defined by a b is a ring homomorphism. a Ф(а + bi)- a

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Q: QUESTION 2 Suppose f:R+ F is a surjective ring homomorphism from a ring R to a field F. Either prove…

A: Given: f:R→F is surjective ring homomorphism from ring R to a field F. To prove: R is field or give…

Q: Let R be a ring. Consider the map Ø:Q[x]→Q defined by Ø(f(x))=f(3 Then, the Kernel of Ø is:

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Q: If µ is finitely additive on a ring R; E, F eR show µ(E) +µ(F) = µ(Eu F)+µ(En F) %3D

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Q: Example (3) :- Let fa:(R, +, .) (R, +, .) be a ring . homo ,where R is a ring with identity s.t.,…

A: We find kernel of given ring homomorphism

Q: Let R = {; la. b e z}and let p:R - Zbe defined by : 0( ) = a 1) is a ring a) Homomorphism. b)…

A: Since you have asked multiple question,as per our guidelines we are supposed to answer only one…

Q: Problem 2 mapping : k[x] → k(a) defined by (p) = p(a) is a ring homomorphism. Let K be an extension…

A: if F=KF=K, then every element a∈Fa∈F can be written as a⋅1a⋅1 with a∈Ka∈K and 1∈F1∈F. Thus {1}{1} is…

Q: QUESTION 10 Let f:R S be a ring homomorphism. (i) Prove that if K is a subring of R then F(K) is a…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 5. Let R be a commutative ring with identity. Let f: R[x] → R be defined by f(p(x)) = Eo ai, where…

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Q: Show that if R, R’, and R’’ are rings, and if ø : R → R’ and ψ : R’ → R’’ are homomorphisms, then…

A: Ring homomorphism: A mapping f: A → B between ring A and B is said to be homomorphism if it…