Let A = {S : S is a subring of C and e S}, and let R = N S be SEF the intersection of all these rings. (a) Give an example of a subring of C that is not in A.

Q: The ring Z pg?, has exactly------------maximal ideals 2 3 1 4

A: An ideal I in Zn  is maximal if and only if I=⟨p⟩ where p is a prime dividing n.

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

A: Subring Test : A non empty subset S of a ring R is a subring if S is closed under subtraction and…

Q: Suppose that K is a commutative ring with identity. If and I are ideals of R for which R/I≈ R/J as…

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Q: Let R be a ring with unity 1. Show that S = {n· 1 | nE Z} is a sub- ring of R.

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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 R be a ring with identity 1 and let a be an element of R such that a2 = 1. Let S = { ara : r e…

A: we will use sub ring test.

Q: 3. Prove that an ideal I in a ring R is the whole ring if and only if 1 e I.

A: Question: Prove that an ideal I in a ring R is the whole ring if and only if 1∈I. Proof: We have to…

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

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Q: Let R be a commutative ring with unity, and let c ER be a fixed element. (a) Prove that the set A =…

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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: 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: 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: Let R be a ring with unity 1. If 1 has infinite order under addition then the characteristic of R is…

A: The order of an element a in a ring R is the least positive integer n such that n·a=0. If no such…

Q: Suppose R is a commutative ring and |R|= 30. If I is an ideal of R and |I| = 10, prove that I is a…

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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: 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: (17) Prove that the ring Zm Xx Z, is not isomorphic to Zmn if m and n are not relatively prime.

A: We have to prove given property:

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

A: The given statement is ,The ring Z8i has no proper ideals.

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: 35. Show that the first ring is not isomorphic to the second. (a) Eand Z © Z × Zu and Z () ZXZ, and…

A: The objective is to show that the first ring of the following is not isomorphic to the second:

Q: The rings Z and 5Z are isomorphic.

A:

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: Q: Let S, and Szare two subrings of a ring (R, +,.), prove that S, US2 is subring of R iff either S,…

A: By supposinɡ S1 and S2 as two subrinɡs of rinɡ (R, +, .) To prove that S1∪ S2 is subrinɡ of R if and…

Q: Let R be a ring and S a subring of R. Prove that if 1R ∈S, then S has unity and 1S =1R

A:

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: Exhibit a commutative ring R and an element x E R such that Z CR and x is NOT prime but irreducible…

A: Take R = Z[i√5]  Clearly, R is commutative ring and Z ⊆ R  Also, 2,3 ∈ R are not prime but…

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: 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: 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: 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: Suppose R is a commutative ring with 1R# 0R. Show that if f (x) = ao + a1a + a2a ++a,n" is a unit in…

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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: The ring Z,3 has exactly------------maximal ideals 3 4 1 O O O O

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

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Q: (d) Let S = (Q, +, ·), the ring where + is addition of rational numbers and · is multiplicaiton of…

A: The given question is related with abstract algebra. We have to solve the following :…

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

A: