Let R be a ring and S be set. Denote by RS the collection of all functions f: S→ R. Equipped RS with the pointwise addition and pointwise multiplication f+g: S → R S → f(s) + g(s) R S → f(s)g(s) fg S → and for any f,g: S→ R € RS. Prove that RS is a ring with respect to the operations define above.

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: Let R be a ring in which each element is idempotent. Let R = R XZ₂. Define + and x on R by (a, [n])…

A: Given: R is a ring in which each element is idempotent and R=R×Z2.To show: + and × are well-defined…

Q: let k. be the splitting field of fax) &F (20) and let poxy be con irreducible factor of few in F(x).…

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

A: Let x, y ∈ R. Then f(x + y) = (x + y)2 = x2 + 2xy + y2 = x2 + y2, since R is commutative and 2x=0…

Q: 2. Consider the following functions. Are these ring homomorphisms? If yes, prove it. If no, provide…

A: a) f:Z→Z given by f(x) = 3x For this function to be a ring homomorphism, it needs to satisfy two…

Q: 4. Let ƒ : Zn → Zm be a ring homomorphism such that f(1) = 1. Show that m divides n.

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Q: Let R and S be rings, F a field, and f : R → S, g : F → R ring homomorphims. (a) Prove that, for any…

A: Sol:- (a) Let r be an element of R and n be an integer. Then we have: f(nr) = f(r + r + ... + r) (n…

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: To submit Let R be a ring. Define a function e : R[x] × R → R by the rule e(a„x" + ·..+a¡x+a0,r) =…

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Q: Let R be a ring with identity, f: RS be an onto homomorphism and r ER be a unit. Show that f(r) ES…

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Q: For each point a∈R of the ring ≺R,+,⋅≻, the mapping θa:≺R,+,⋅≻→≺R,+,⋅≻ defined by θa(x)=a⋅x,  ∀x∈R…

A: Consider R = Z6 Then, we know obviously that (R, +, . ) is a ring. Consider a = 2.So,  θ2(x)=2x

Q: Denote by Zm = {[0]m, [1]m, ..., [m1]m} the ring of integers modulo m. Consider the rings R = Z24…

A: To decide whether a given map θ is  an isomorphism of earrings, we should take a look at if it…

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: Denote by Zm = {[0]m [1]m,..., [m - 1]m} the ring of integers modulo m. Consider the rings R = Z24…

A: Hi! Please refer to the attachment. Thank you!Explanation:

Q: sider the integers Z as a Z-module. Let n be a non-zero integer and define a function fn :ZZ by…

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Q: (b) (-a)b = −(ab) = a(−b) for all a,b € R.

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Q: Let R be the ring Z[V3] = {a+b/3|a, b E Z}. For each of the following elements of R determine if it…

A: let R be the ring ℤ3=a+b3|a,b∈ℤ claim- for each element of R determine if it is an invertible…

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: Consider ϕ: ℤ × ℤ → ℤ defined by ϕ(a,b) = a-b. Given that ϕ is a homomorphism, find its kernel and…

A: Let ϕ:*(Z*Z)->Z is a function. Then ϕ is a homomorphism iff ϕ(a+b)=ϕ(a)+ϕ(b) for all a,b ∈Z*Z

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: Let R be a ring and f : R → R be defined by f(x) = x4. Check All that are correct. O fis not onto…

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Q: 2. Let R be a ring. The center of R is the set {x = R | ax = xa for all a € R}. Prove that the…

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

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

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Q: Suppose R is commutative. Prove that the map oa : R[x] → R where $₁(p(x)) = p(a) for some a € R is a…

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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: Determine whether the following mapping is a ring homomorphism h: Z[V2]→ Z[v2] defined by h(a +…

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Q: If f is a ring homomorphism from Z„ to Z„ such that f (1) = b, then b4*+2 = b*. O True False

A: Given f : Zm→Zn is a ring homomorphism such that f1=b Here, we have to check whether b4k+2=bk is…

Q: Let f: R → S be a ring homomorphism and T ≤ S. Prove that K = { r ∈ R : f (r) = 0s} and P = { r ∈ R…

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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: Let ø : F → R be a ring homomorphism where F is a field. Show that either o is one-to-one or ø is…

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Q: 6. Let R be a commutative ring with identity and let f be an onto homomorphism from R to a ring S.…

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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 (fg)(x + y) = f(g(x + y)) = f(g(x) + g(y)) = f(g(x)) + f(g(y)) =(fg)(x)+(fg)(y),…

Q: Let T be the set of all continuous functions from R to R. 1) Is T a commutative ring with identity?…

A: This question is about application number theorem

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

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Q: Recall that C(R) is the ring of continuous real-valued functions on R. Define φ : C(R) → R by φ(f )…

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Q: Let x, y, and z be an element of a Ring Prove that x(y-z) = xy - xz

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Q: Let K = ℚ(sqrt2, i ). Find all isomorphisms of K onto a subfield ℚ(bar) as extensions of the…

A: Let K=Q2,i Let ϕ:K→K, So ϕ depends on Q2 and Qi. Now, minimal polynomial of 2 is x2−2 So, other…

Q: (9) Let R be a ring such that x² = x for any x ЄR. (a) Prove that it is commutative. (b) Can R be an…

A: Let R be a ring such that x2=x for any x∈R. (a) Let a,b∈R. Then a2=a, b2=b and (a+b)2=a+b. Since…

Q: Let f: Z6→ Z2 × Z3 be defined by the following specification of its values 0 → (0, 0) , 1 → (1, 1) ,…

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Q: 7. Let R be a ring, S a subring of R. We define a relation ~ on R by x ~ y if and only if x – y E S.…

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Q: Let : A B be a mapping of sets, A a system of subsets of A, and 3 a system of subsets of B. Define…

A: Since you have asked multiple questions, we will solve the Question (k) and (l) for you. If you want…

Q: 6 Define f:Z- 42 by f(x) - 4x, for all x €Z. Then: (a) f is a ring epimorphism. (b) / is a ring…

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Q: 8. For f(z) = e², find arg f'(z). Use this to verify that lines parallel to the y axis and x axis…

A: Given: f(z)=ez To find: arg(f'(z)) To verify that lines parallel to y-axis and x-axis are mapped…