Indicate such a subring of the ring P[x] that contains P and is different from P but is not isomorphic to P[x].

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: IF J is nil left ideal in an Artinian ring R, then J is

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Q: Let K be a commutative ring. For every AÃ ÃA = det (A) In- = quence, A is invertible iff det(A) is…

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Q: Let ø : R  → R’  be a ring homomorphism and let N be an ideal of R.  Let N’ be an ideal either of ø…

A: Ideal: A non-empty subset I of a ring R is said to be ideal in a ring I if it satiesfies following…

Q: Let R be a commutative ring with unity, and let I be a proper idealwith the property that every…

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

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Q: Suppose that Φ: R --> S is a ring homomorphism and that the imageof Φ is not {0}. If R has a…

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Q: Find an example of a commutative ring A that contains a subset, say S, such that for every s E S we…

A: We can find n commutative ring A such that A contains a subset S such that for every s in S we have…

Q: Let n be a positive integer. Show that there is a ring isomorphismfrom Z2 to a subring of Z2n if and…

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Q: Let R be a ring with unity 1 and char (R) = 3. Then R contains a subring isomorphic to

A: Let R be a ring with unity 1 and char(R)=3.  Then R contains a subring isomorphic to_______.

Q: Let R be a ring with unity 1 and char (R) = 4. Then R contains a subring isomorphic to

A: Let R be a ring with unity 1 and char(R)=4.Then R contains a subring isomorphic to________

Q: Let R be a commutative ring with 10. Prove that R is a field if and only if 0 is a maximal ideal.

A: If R is a field, then prove that {0} is a maximal ideal. Suppose that R is a field and let I be a…

Q: Let F be a field. Show that in F[x] a prime ideal is a maximal ideal.

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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: 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: If R is a commutative ring, show that the characteristic of R[x] is thesame as the characteristic of…

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Q: Let R be a commutative ring. If I and P are ideals of R with P prime such that I ¢ P, prove that the…

A: The ideal quotient of P and I is P:I=x∈R : xI⊂P which is again an ideal of R. Given that P is a…

Q: Prove that a ring with unity R has a unique maximal left ideal M if and only if R\M is the set of…

A: Prove that a ring with unity R has a unique maximal left ideal M if and only if R\M is the set of…

Q: Show that the centre of a ring R is a sub ring of R. And also show that the centre of a division…

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Q: - Let R be a commutative ring with unity of prime characteristic p. Show that the map op : R →R…

A: Homomorphic ring means F:R---->R' and a,  b belongs to R then f(a+b) = f(a)+(b)        and f(ab)…

Q: Show that if R is a ring with unity and N is an ideal of R such that N R, then R/N is a ring with…

A: Since we have given that R is a ring with unity and N is a proper  ideal of R . We prove R/N is a…

Q: Is it true that if S is a unital subring of a unital ring, then the identity elements of the two…

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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: 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: Let R be a commutative ring with identity. Is x an irreducible element of R[x]? Either prove that it…

A: Given that R is a commutative ring with identity.

Q: Is the ring Z/6Z and Z6 are isomorphic.

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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: Suppose phi maps R to S is a ring homomorphism and the image of of phi is not {0}. If R has a unity…

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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: The rings Z and 5Z are isomorphic.

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Q: Let f : R→ R' be a ring homomorphism of commutative rings R and R'. Show that if the ideal P is a…

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Q: Let f : R S be a homomorphism of rings, 1. If K is a subringof R, Is o(K) a subring of S? If so,…

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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: a) If U and V are ideals of a ring R and let UV be the set of all those elements which can be…

A: We are given that U and V are the ideals of a ring R. Now, we need to show that UV is an ideal of R.…

Q: Find all ring homomorphisms from R to Z5.

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Q: Show that if R is a ring with unity and N is an ideal of R such that N ≠ R , then R / N is a ring…

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Q: elements

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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: 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: 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: a. Let R and S be commutative rings with unities and f:R -S be an epimorphism of rings. Prove that S…

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Q: Suppose that R is a commutative ring with unity and that I is an ideal of R. Prove that the set of…

A: Given: R is a commutative ring with unity and that I is an ideal of R. To prove: The set of all x∈R…

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

A: Image is attached with detailed solution.

Q: Prove that the intersection of any collection of subrings of a ring Ris a subring of R.

A: Let S be  intersection of any collection of subrings of ring R. Then we have to prove  S is subring…

Q: 2) Let P + Q be maximal ideals in a ring R and a,b elements of R. Show that there exists c E R, such…

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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: (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: 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: The ring Zp2, has exactly-----------maximal ideals

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