Determine all reversible elements in the rings P[x] and 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: The function p: Zs → Z30 defined by ø(a) = 6a is a ring homomorphism. True False

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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: 4) Let f : C – C be a homomorphism of rings with f(x) = x for all x E R. Show that f is either the…

A: If a function f is ring homomorphism from a ring R to another ring R'. Then We knew from the…

Q: Find the characteristic of each of the following rings   E

A: We need to find the characteristics of the ring E. If there are positive integers n such that nx=0…

Q: 12. Prove that the rings Z,[x]/(x² + x + 2) and Z,[x]/(x² + 2x + 2) are isomorphic.

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

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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: The ring 5Z is isomorphic to the ring 6Z OTrue O False

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Q: Suppose that a belongs to a ring and a4 = a2. Prove that a2n = a2 forall n >= 1.

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Q: Find the degree of the splitting field of x^2-3 over Q.

A: Degree of splitting field

Q: Let K/L be ideal of quotient ring R/L. If K is prime ideal and contains L, then K/L is prime ideal.…

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Q: Let T be ring containing elements e, f, both # 0T, such that e + f = 1r , e² = e, f² = f , and e · f…

A: We are given e+f = 1T,e2=e,f2=f and e.f=0T where T is the rings containing elements e,f both not…

Q: Suppose that F is a field and there is a ring homomorphism from Zonto F. Show that F is isomorphic…

A: F is a field. Consider φ as a ring homomorphism from Z to F. As φ is onto. Thus φ(Z) = F.

Q: There are.... Polynomials of degree atmost n in the polynomial ring Z, [x]. 5^n O 5+ 5^n Onone…

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Q: Any ring R is a Jacobsen radical ring, if it is a simple ring. O مبا O

A: JACOBSON RADICAL:- The radical of the base ring R is called its Jacobson radical and denoted by…

Q: 4 If R is a commutativ ring, show that the characteristic of R[X] is the same the characteristic of…

A: Given

Q: The measure u is monotone on the ring. So that µ(A) < µ(B) if ACB

A: Given that The measure is monotone ob the ring,  So we need to consider the following;

Q: Let a be an element of a ring R such that a3=1R. Prove: for any integer n, either (an)n=1R or…

A: Let a be an element of a ring R such that a3=1R.  We will find, for any integer n, either (an)n is,…

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: The ring Z75 has exactly-----------maximal ideals 4 O 2 6 3

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Q: The ring Z3[i] has no proper ideals O True O False

A: Yes it is true .

Q: Let R be a ring with m elements. Show that the characteristic of Rdivides m.

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Q: If p = 1 mod 4, then we can still define F,[i] as a ring, but it is not a field. Illustrate this in…

A: Consider a = 1 + 2i in F5[i] and b = 2 + i in F5[i] Then, a and b both are non-zero element in…

Q: Q::Let S1 and S2are two subrings of a ring (R, +,.), prove that S, U S2 is subring of R iff either…

A: 1 Let S1 and S2be two subrings of R,+,.. First suppose that either  S1⊆S2 or S2⊆S1 we will prove…

Q: * If K is a simple ring, then K a Jacobsen radical ring. T F

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

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Q: 30. Let R be a ring with identity lr and S a subring of R with identity 1s. Prove or disprove that…

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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 F be a field and let S be a ring with |S| > 1. Let p: F → S be a non- zero ring homomorphism.…

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Q: Find all possible ring homomorphisms from phi: Z[i] to Z[i]

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Q: Q4: Define ring. Is every subset of a ring R also a ring? Is a a ilnotent elen

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Q: Show that if n is an integer and a is an element from a ring, thenn . (-a) = -(n . a).

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Q: Find all ring homomorphisms from R to Z5.

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Q: 18. Let p:C + C be an isomorphism of rings such that e(a) - a for each ae Q. Suppose r E Cisa root…

A: Given: Let ϕ:ℂ→ℂ be an isomorphism or rings such that ϕ(a)=a for each a∈ℚ. Suppose r∈ℂ is the root…

Q: in a commutative ring with unit element, prove that the relation "a is an associate of b" is an…

A: Reflexive: As "a is an associate of a '' so, (a,a) belongs to the relation. Therefore, given…

Q: The number of nilpotent elements in the ring Z23 is:

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Q: For any n E N, each zero divisor in the ring Z, is nilpotent.

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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: Suppose I, J be ideals of a commutative ring R. Prove that IJ cIn).

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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: The ring Zpg?, has exactly-------------maximal ideals O 2

A: 3

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

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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: The ring Z-[i] has no proper ideals

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

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Q: Let A be a ring and suppose that for some even positive integer n we have a" =a. Then, for every a E…

A: