7. a) Let R be an integral domain and a € R. Show that the function f : R[x] → R given by f(p(x)) = p(a) is a ring homomorphism. This is often called the evaluation homomorphism at a.. b) Find Ker f. It has a nice description in words.

Q: (b) Let R, S be rings and f: R→ S be a map defined by f(r) = Os for every r ER, where OR, Os are the…

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

A: Introduction: Like group homomorphism, ring homomorphism also exists. A ring homomorphism is a…

Q: Let ideal МоӨ by (H₂O)(x) = Sup { min{min; M(Y;), 0(2:17}]} x = [² y ₁ Z ₂² finite where y., 2. ER…

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Q: Let I = = {[x] x, y = R} and J = {[2] ZER} E Consider the ring homomorphism y: I → R defined as (a)…

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Q: {[X] x, y € Z} and the mapping of f: H → Z y given that f(x) Consider H = {[ = x - y.

A: Given that H=xyyx |x,y∈ℤ. The map f:H→ℤ defined by fxyyx=x-y. We have to prove that f is a ring…

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: (3) (D) Let R be the ring of all functions from R to R. Let I be the subset consisting of those…

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Q: 1. Let T be a group of all invertible 2 x2 matrices of the form Where a.b.C ER and ac+0. Let / be…

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Q: Let A be an associative algebra. (a) Show that for any a E A the map da : A A defined by da(b) = ab…

A: Let A be an associated algebra.  To prove that for any a∈A, the map da:A→A defined by dab=ab-ba is a…

Q: 4. Let R be a ring. (a) Let x, y, z € R. Prove that, if x | y and y | z, then x | z. (b) Prove that…

A: Let R be  a  ring .4) a)  Let  Suppose that   divides y and  y  divides z.We  have  to show that x…

Q: -se A, B are submodules (that is, they are isom : M/A→M/B and show pism, then use the FHT

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

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Q: Let R=Z and R'= set of all even integers. Then %3D (R', +, *) is a ring, where a* b= ab V a, be R'.…

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Q: = (19) Let om : Z → Zm be the ring homomorphism defined by σm(a) = remainder of division of a by m.…

A: Ans-19)We are given in the question (19) Let om Z→ Zm be the ring homomorphism defined by om (a) = =…

Q: Let T: R² R² be given by T(x1, x₂) = -[at] It is known that I is an isomorphism. T-¹ is given by the…

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Q: 2. Let s-{(소)아 2a S = 26 2a Assume that S is a subring of M2(Q). Prove that S is isomorphic to Q[V2]…

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Q: Problem 1. Let R be a ring. Construct an isomorphism between¹ R[X,Y]/(XY) and the subring A CR[X] ×…

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Q: Let R be the ring of all continuous real valued functions on the closed interval [0, 1]. Prove that…

A: Here R be the ring of all continuous real valued functions on the closed interval [0,1] We have to…

Q: Exercise 7.24. (a) Suppose : Z36 → G is a surjective homomorphism, where |G| (b) (a) Suppose : Z25 →…

A: (a) Here we have given that ϕ:ℤ36→G is a surjective homomorphism, where G=12. Now, by the first…

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…

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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: 3- Let a: (F(R), +..) → (R, +..) be the map defined by a(f) = f(3)-f(0). Then o is a ring…

A: Given: σ:Fℝ, +, ·→ℝ, +, · be the mapping defined by, σf=f(3)-f(0) To determine: Whether σ is a ring…

Q: M n of ring (5 Marks) (b) Let (R,+,) and (R', +', '') be any two rings, let f: RR' be a ring…

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Q: (a) Let R and R' be two rings and : R→ R' a map. Define what it means for to be a ring homomorphism.…

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Q: 2. If f: R→ S is a homomorphism of rings and u is a unit in f, prove that f(u) is a unit in S

A: We recall the definition of a unit and then try to solve this problem. It is the definition of an…

Q: Problem 3 Let R be any ring with ideals I and J such that J CI Let 1/] = {a +J| a e 1} Prove…

A: NOTE:- AS POSTED MULTIPLE PART OF THE QUESTION, I WILL ANSWER ONLY FIRST THREE PART AS PER BARTLEBY…

Q: Let p: Z → Z30 be the map p(x) = 21x. 1. Show that p is a ring homomorphism. 2. Find Ker(y). 3. Find…

A: Definition of Homomorphism:        It is a structure preserving map between two algtebraic…

Q: 11. Let R and R' be two rings. A mapping f: R→R' is called an antihomomorphism, if f(x+y)=f(x) +…

A: Let x, y ∈ R. Then (g)(x + y) = f(g(x + y)) = f(g(x) + g(y)) = f(g(x)) + f(g(y)) =(fg)(x)+(fg)(y),…

Q: Suppose (X,≺) is a well-ordering and f : X → X is an isomorphism. Then f is the identity function on…

A: Well ordering

Q: Exercise: Prove that the homomorphism f:R R of the ring R onto the ring R' is isomorphism if and…

A: We have given that , f : R → R' be homomorphism. Also we have given that ,  f : R → R'  is onto. We…

Q: 2.3 Let 9: (M2(R), +)→ be defined by e c d a b = a +d. where a,b,c,d €R (a) Prove that e is a…

A: A mapping f:(G,+)→(H,+) is said to be group homomorphism if it satisfies: f(a+b)=f(a)+f(b)…

Q: 15. Let R be a ring, I, J be ideals of R and f: RR/I x R/J be the function defined by f(a) = (a +…

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Q: Let p: Q[x] → M2(R) be defined by p(ao + a1 + a2x² + ...+ anx") = ( ) ao (1) Write down o(2 +x –…

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Q: Check whether the function f: A → B is or is not a homomorphism of ring A in ring B in the following…

A: Given  f:A→BA=ℤ & B=ℤ such that (a) fx=x+1 (b) fx=2x We need to see if f:A→B is homomorphism of…

Q: Let φ: ℤ→ℤ defined by φ(x) = x + 2. Show φ is a homomorphism.

A: We must demonstrate two characteristics, preservation of addition and preservation of identity, in…

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

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