Label each of the following statements as either true or false.   A ring homomorphism from a ring R to a ring R' must preserve both ring operations

Q: (a) Let R be a ring and S a subset of R. What does it mean to say that S is a subring of R?

A: a. S is a subset of R. A non-empty subset S of R is a subring if a, b ∈ S ⇒ a - b, ab ∈ S. A subring…

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: Give an example of a subring of a ring, say A, that is not an ideal of A.

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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: Label each of the following statements as either true or false where R represents a commutative ring…

A: The statement " The degree of the product of any two polynomials f(x) and g(x) over an domain R…

Q: If S is a subring of a ring R, then S[a] is a subring of R[x]. Exercise 2.35.1 Prove this assertion!…

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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: Let R be a ring with a subring S. Prove or disprove: If a ∈ R is nilpotent and a ∈ S, then a is also…

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Q: Indicate such a subring of the ring P[x] that contains P and is different from P but is not…

A: There are so many examples can be found.

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

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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: a is a unit in a ring R with unity, then a is not a zero divisor

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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: If R is a ring, then every element of R is either a unit or a zero-divisor. Select one: O True O…

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Q: The ring Z is isomorphic to the ring 3Z False True

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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: 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: 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: Show that each homomorphism from a field to a ring is either one to one or maps everything onto 0.

A: Let ϕ:F→R  be a ring homomorphism from the field F to ring R . Now, the  kernel of ϕ is ideal of F.…

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

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Q: Let I be the set of all elements of a ring R that have finite additive order. Prove that I is an…

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Q: Let R be a ring. Prove that the set S = x R / xa = ax, a Ris a subring of R .

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Q: Let R be a ring with a subring S: Prove or disprove: If a ∈ R is a unit, and a ∈ S, then a is also a…

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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 ring and a=a for all a'e R, Then commutative. prove that R is

A: First we notice that x3=x for all x∈ℝ, so that means 2x3=2x and thus 8x=8x3=2x and so 6x=0. Thus…

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

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

A: .

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: 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: Label each of the following statements as either true or false.   Every ideal of a ring R is a…

A: Every ideal of a ring R is a subring of R

Q: If I is an ideal of a ring R, prove that I[x] is an ideal of R[x].

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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: Label each of the following statements as either true or false. .Let R be a ring with unity and Sa…

A: We have given a ring R with unity and S a subring (with unity) of R. We have to tell given statement…

Q: If R1 and R2 are subrings of the ring R, prove that R1 n R2 is a subring of R.

A: R1 and R2 are subrings of the ring R, prove that R1∩R2 is a subring of R

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 R be a commutative ring with unit element .if f(x) is a prime ideal of R[x] then show that R is…

A: Given R be a commutative ring with unit element. If f(x) is a prime ideal of R[x] then we have to…

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: Prove that Q[x]/<x2 - 2> is ring-isomorphic to Q[√2] = {a +b√2 | a, b ∈ Q}.

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Q: Label each of the following statements as either true or false. Any ring with unity must be…

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Q: Show such a subring of the ring P[x], which contains P and is different from P and is not isomorphic…

A: The given details: The ring is P[x]. To show that the subring of the ring P[x] which contains P and…

Q: Label each of the following statements as either true or false.   The only ideal of a ring R that…

A: The ring R is defined as a set that is not empty and contains a minimum of 1 element in it. This…

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

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Q: Construct a non-trivial Cayley's table of a ring isomorphic to the ring Z,. Also write the mapping…

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Q: Label each of the following statements as either true or false. The characteristic of a ring R is…

A: To determine whether true or falseThe characteristic of a ring R is the positive integer n such that…

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: Label each of the following statements as either true or false. Let R be a ring. The set { 0} is a…

A: State: True Or FalseLet R be a ring.The set 0 is a subring of R with respect to the operations in R.

Q: Let R be a ring with unity e. Verify that the mapping θ: Z---------- R defined by θ (x) = x • e is a…

A: Let R be a ring with unity e, verify the mapping θ:Z→R defined by θx=x.e is a homomorphism If R and…

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…