Let A, B,C be rings. Let & be a ring homomorphism from A into B and ß be a ring homomorphism from B into C. Prove that • 4 : 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 = {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: Let R be a commutative ring. Then R is an integral domain if and only if ab= ac implies that b = c,…

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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: If Ø: R S is a ring homomorphism. The Ø preserves:

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Q: Suppose that R and S are commutative rings with unities. Let ø be a ring homomorphism from R onto S…

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Q: 4. Let y: R→ S be a ring homomorphism. Prove that ': R[X] → S[X] given by soʻ (ao + a1X + ..a,X") =…

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

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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: If R is a commutative ring with unity and a e R, then (a) = {ar : reR}=aR.

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Q: consider the mapping from M2(z) into Z,prove or disprove that this is a ring homomorphism

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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: 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: 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 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: 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: 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: let (Z,+,*) be a ring of integer number and (Ze,+,*) is ring of even integer number and f:Z→Ze such…

A: Given : (Z,+,*) is a ring of integer numbers. (Ze,+,*) is a ring of even integer numbers.   To…

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: 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: 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: Let R be a commutative ring of characteristic 2. Prove that : (a+ b) = a² +b² = (a - b)? v a, be R.…

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Q: The rings Z and 5Z are isomorphic.

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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: Q2) Let(M₂ (R), +..) be a ring. Prove H = {(a) la, b, c = R}is a subring of (M₂ (R), +,.).

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Q: Let S[x] = {a+bx where a,b in ℝ} (S[X] is ring)  and  ℝ.  Prove that there is no isomorphism between…

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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 unity…

A: In the given question we have to tell that " Is f(x)=6x is a ring homomorphism from (Z5,⊕5,⊗5) to…

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: Find all ring homomorphisms from R to Z5.

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

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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: Iffis a ring homomorphism from Zm to Zn such that f(1) = b, then b4k+2 = bk. True False

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Q: Let A = {S : S is a subring of C and e S}, and let R = N S be SEF the intersection of all these…

A: Given that A=S| S is a subring of ℂ and 12∈S and let R=∩S∈AS. (a) Consider the ring M=3k: k∈ℤ.…

Q: 1. Suppose that p: R → S is a ring homomorphism and that the image of o is not {0}. If R has a unity…

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

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

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

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Q: (b) Show that if : R→ S is only assumed to be a ring homomorphism, then it is possible to have a…

A: We can prove the above result using two contradictory examples. Note that these examples are only…

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…