Prove that if Ø is a ring homomorphism from Z, to itself of the form Øx) = ax,then show %3D that a2 =a (mod n).

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: if in a ring R every x in R satisfies x^2=x , prove that R must be commutative

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Q: if A and B are ideals in a ring R such that A intersect B ={0}, prove that for every a in A and b in…

A: Let A and B are ideals of a ring R such that A∩B=0

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

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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: 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: As a ring, z is isormorphic to nZ for all n>1. True False

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

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Q: - Prove that, if I is an ideai of the ring Z of integer numbers then I=, for some nɛZ'U{0}

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

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Q: Iffis a ring homomorphism from Zm to Zn such thatf(1) = b, then bak+2 = bk. True False

A: Ring homorphism

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: If in a ring R every x E R satisfies x2 = x, Prove that R must be commutative.

A: Answer and explanation is given below...

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: determine all ring homomorphism from Z20 to Z30

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Q: If Ø: R→ S is a ring isomorphism. The Ø preserves: O All of these O Nilpotent elements O Idempotent…

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Q: 5. An element x in a ring R is called idempotent if a2 = x. Prove that if a is an idempotent element…

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Q: Prove that Rn+3/Rn and R3 are isomorphic rings for all n belonging to N

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Q: If R is a commutative ring with unity and a e R, then (a) = {ar:reR}=aR. %3D %3D

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Q: Let R be a commutative ring with identity. Using the homomorphism theorem (Theorem 16.45) and…

A: Recall that in a ring A  not necessarily commutative and with an identity, an ideal M⊂A is a maximal…

Q: Show that the subring of an artinain ring may not be artinian???

A: According to the given information, it is required to show that subring of an Artinian ring may not…

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: A ring, R, in which r^2 = r for all elements r in R is called a Boolean ring. In any Boolean ring,…

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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: 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: Q1: Let S, and Szare two subrings of a ring (R, +,.), prove that S, USz is subring of R iff either…

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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: Iffis a ring homomorphism from Zm to Zn such thatƒ(1) = b, then b4k+2 = bk. %3D True False

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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: 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: Prove or dlspive that the subset M: ta+3biabeZ5 IS a subring of the Causslan integer ring ZCiJ

A: If R is a ring  and S is a subset of R is said to be a  subring if it is closed in the following…

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: Let be a commutative ring with unity of characteristic 3. Compute and simplify (a + b)6 for all…

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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: Find all ring homomorphisms from R to Z5.

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

A: 3

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: Prove that if (I,+,.) is an ideal of the Ring (R,+,.) then rad I= In rad R ???

A: Solution :

Q: 2) Let M be a noetherian R-module over commutative ring R. If f : M M is onto ho- momorphism , show…

A: Please check step 2 and 3 for solution.

Q: Let R be a ring such that a6 - = x for all æ E R. Prove that R is commutative.

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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: (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.