Let A, B,C be rings. Let o be a ring homomorphism from A into B and B be a ring homomorphism from B into C. Prove that • : A → C is a ring homomorphism.

Q: Construct a homomorphism of rings p:Z[i] → Z,

A: Consider the rings ℤi and ℤ2. Define a map φ:ℤi→ℤ by φa+ib=0 ∀ a,b∈ℤ. Let a+ib, c+id∈ℤi.…

Q: Let ø : R → S be a ring homomorphism, and define ø* : R[x] → S[r] as follows. For f(r) = ana" + ...+…

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Q: Let R be a ring with unity 1 and char (R) = 4. %3D Then R contains a subring isomorphic to Q ZO Z3 O

A: IN the given question,  Given that: R is a ring with unity 1 and char(R)=4. we have to find: we have…

Q: Let R = {2n: n E Z} and define addition O and multiplication O in R by a.b a O b : = a + b and aOb…

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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: Let R be a commutative ring and let x, y ∈ R. If xy is a zero divisor, show that x or y is a zero…

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Q: Let R be a commutative ring. Prove that HO.R (R, M) and M are isomorphic R-modules.

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Q: Prove that the only homomorphisms from Z to Z (Z being the ring of integers) are the identity and…

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Q: Iffis a ring homomorphism from Zm to Zn such that f (1) = b, then bak+2 = bk. True O False

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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: Iffis a ring homomorphism from Zm to Zn such thatf(1) = b, then bak+2 = bk. True False

A: Ring homorphism

Q: If R is a commutative ring and Ø:R→S is a ring homomorphism, then S is a commutative ring True O…

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Q: Let R be a ring such that a² = a for all a E R, and assume that R has an identity. Show that the…

A: Given R be a ring with unity 1 such that a2=a, for all a∈R. Let b∈R be a unit in R. Therefore b-1…

Q: Determine all ring homomorphism from Zn to Zn

A: (if n is not prime) No. of ring homomorphism from Zn to Zn = No. of Idempotent elements in Zn.

Q: Determine all ring homomorphisms from Z to Z.

A: In algebra, A homomorphism is defined as the similarity between the shape, structure, form, etc…

Q: Let R = {2n: n E Z} and define addition and multiplication O in R by a b = a + b and aOb = for all…

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Q: determine all ring homomorphism from Z20 to Z30

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Q: 1- Let ý:R, » R, be a ring homomorphism such that Kerø =. Then, o is a) 1-1 b) onto c) Both 1-1 and…

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Q: Let U = {a, b}. Define addition and multiplication in P(U) by C +D = CU D and CD = Cn D. Decide…

A: Ring (definition) Let R be a non empty set together with two binary operations called addition(+)…

Q: Let R and S be rings and let ø : R → S be a ring homomorphism. Show that o is one-to-one if and only…

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Q: Let f:R, » R, and let g : R, » R, be two ring isomorphisms. If ø =g•f, then 2- ø(e,) = a) , b) € c)…

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Q: For a fixed element a of a ring R, prove that the set {x ϵ R I  ax = O} is a subring of R.

A: Given :            The ring R and a fixed element 'a' of R.   To prove that the set x ∈ R | ax =…

Q: 1- Let ý:R, » R, be a ring homomorphism such that Kerø = . Then, ø is a) 1-1 b) onto c) Both 1-I and…

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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 ø is a ring homomorphism from R to S. Then i. ii. Prove that (kero) is an ideal of S. Prove that…

A: Given φ is a ring homomorphism from R to S. To prove: φkerφ is an ideal of S. Given, φ: R→S is a…

Q: Suppose that : R → S is a ring isomorphism. Then g-l is a ring homo- morphism, and hence a ring…

A: According to the given information:

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 R, , O is a ring under two composion e and O üs follows ü e i; = a + b + 1 and aOb = ab + a + b…

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Q: 7. The number of distinct homomorphisms from the ring (Z, +,.) onto the ring (Ze, +,.) is (a) 0 (b)…

A: Number of distinct onto homomorphism is

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 R be a commutative ring. Show that R[x] has a subring isomorphicto R.

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Q: Is the mapping from Z5 to Z30 given by x --> 6x a ring homomorphism?Note that the image of the…

A: Given a mapping,

Q: Let (R, +, .) be a nontrivial ring with * identity, prove that 1 0

A: It is given that (R,+, .) be a nontrivial ring with identity. Now we have to show that 1≠0. So, (R,…

Q: Iffis a ring homomorphism from Zm to Zn such thatƒ(1) = b, then b4k+2 = bk. %3D True False

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Q: Let A = {" :a, b e Z} and let P: A → Z such that P(z) = a – b for every z e A. Prove that O is a…

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Q: Let R be a ring and assume a∈R is not a zero divisor.Prove that if ba=ca,  then b=c.

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Q: et R be a ring and assume a∈R is not a zero divisor. If ab=ac for some b,c∈R, then b=c

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Q: Iff is a ring homomorphism from Zm to Zn such thatf(1) = b, then bak+2 = bk . O True O False

A: False

Q: Let R be a commutative ring with identity, and let a, b E R. Assume ab is a unit in R. Do a and b…

A: Here given R is a commutative ring with identity. and let a,b∈R and assume ab is a unit. we have to…

Q: Find all ring homomorphisms from R to Z5.

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Q: Let A, B,C be rings. Let & be a ring homomorphism from A into B and ß be a ring homomorphism from B…

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Q: If R is a commutative ring and Ø:R→S is a ring homomorphism, then S is a commutative ring * O True O…

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Q: The center of a ring R is {z ∈ R : zr = rz for all r ∈ R}, i.e. the set of all elements which…

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Q: Let A be a commutative ring with identity and D be an integral domain. Suppose that p: A → D is a…

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

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Q: Let I be a maximal proper ideal of commutative ring with identity R. Prove that R/I is a field.

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Q: Suppose that R and S are isomorphic rings. Prove that R[r] = S[r].

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Q: If Ø: R → S is a ring isomorphism. The Ø preserves: Units Idempotent elements

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Q: Show that a ring R is commutative if and only it a - b = (a+ b) (a - b) for all a, be R.

A: Proof. Let R be commutative. Then ab = ba for all a,b ∈ R.

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…