Iffis a ring homomorphism from Zm to Zn such that f(1) = b, then b4k+2 = bk. True False

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: Let R be a ring with unity 1 and char (R) = 4. %3D Then R contains a subring isomorphic to Q ZO Z3 O

A: IN the given question,  Given that: R is a ring with unity 1 and char(R)=4. we have to find: we have…

Q: If fis a ringhomomorphism from Zm to Zn such thatf(1) = b, then bak+2 = bk. True False

A: given a ring homomorphism.

Q: The function p: Zs → Z30 defined by ø(a) = 6a is a ring homomorphism. True False

A:

Q: Let R be a commutative ring and let x, y ∈ R. If xy is a zero divisor, show that x or y is a zero…

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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 fis a ring homomorphism from Zm to Zn such that f (1) = b, then b4k+2 = bk True False

A:

Q: Let S be a ring. Determine whether S is commutative if it has the following property: whenever æy =…

A:

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

A:

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

A:

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

A:

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

A:

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: If fis a ring homomorphism from Zm to Z, such that f(1)= b, then b**2 = b*. True False

A:

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

A:

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

A:

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

A: Ring homorphism

Q: If R is a commutative ring and Ø:R→S is a ring homomorphism, then S is a commutative ring True O…

A:

Q: If fis a ring homomorphism from Zm to Z, such that f (1)= b, then b4k+2 = bk. %3D O True O False

A:

Q: determine all ring homomorphism from Z20 to Z30

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Q: 1- Let ý:R, » R, be a ring homomorphism such that Kerø =. Then, o is a) 1-1 b) onto c) Both 1-1 and…

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Q: Let R and S be rings and let ø : R → S be a ring homomorphism. Show that o is one-to-one if and only…

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

A:

Q: - The mapping f: Z → Z defined by f (n) = 5n is a ring homomorphism. a) True b) False

A:

Q: 1- Let ý:R, » R, be a ring homomorphism such that Kerø = . Then, ø is a) 1-1 b) onto c) Both 1-I and…

A:

Q: Q1: Let R be a commutative ring with Char(R) = 2 and let p:R → R be defined such that o (a) = a².…

A:

Q: Suppose that : R → S is a ring isomorphism. Then g-l is a ring homo- morphism, and hence a ring…

A: According to the given information:

Q: Let S be a ring. Determine whether S is commutative if it has the following property: whenever ry =…

A:

Q: Let f:R, → R, and let g : R, → R, be two ring isomorphisms. If ø = g• f , then 2- (e2) =

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: 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: If fis a ring homomorphism from Zm to Z, such that f (1) = b, then b*+2 = b*. True False

A:

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

A:

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

A:

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 mapping f:Z → Z defined by f (n) = 5n is a ring homomorphism. a) True b) False

A:

Q: 1. let S: :1 xg*Z a) Show that is a ning. ning. b) Show that ¢: S Z olefined by $ ( [Š 87)-* is a…

A: Solution of part(a): It is given that S=x0y0, x,y∈ℤ. We know that the sum and product of two 2×2…

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: R→s be a ring homomorphism. ve that if K is a subring of then R f(K) is a subring of S ve that c is…

A: Given, if f:R→S be a ring homomorphism. (i) To prove that if K is a subring of R then f(K) is a…

Q: Iff is a ring homomorphism from Zm to Zn such that f (1)=b, then b*+2 = b*. False True

A:

Q: The map f: Z→ Z,o given by f(x)= 2x is a ring homomorphism. Select one. True False

A: SINCE YOU HAVE ASKED MULTIPLE QUESTIONS IN SINGLE REQUEST, WE WILL BE ANSWERING ONLY THE FIRST…

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

A:

Q: 4 The correspondence ø:x > 5x from Z, to Z, is a ring homomorphism. a) True b) False O a) True O b)…

A:

Q: If Ø: R → S is a ring isomorphism. The Ø preserves: Units Idempotent elements

A:

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.

Q: If Ø: R → S is a ring homomorphism. The Ø preserves: Units All of these Nilpotent elements…

A: The units in a ring are those elements which have an inverse under multiplication. An element x of…