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

Q: Let R be a euclidean ring and a,b are non zero elements of R.if a and b are associates,show that…

A: Let R be a Euclidean domain with measure d. Also given that a and b are associates in R. Then from…

Q: Let R = {2n: n E Z} and define addition O and multiplication O in R by a.b a O b : = a + b and aOb…

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Q: (a) Let R be a ring and S a subset of R. What does it mean to say that S is a subring of R?

A: a. S is a subset of R. A non-empty subset S of R is a subring if a, b ∈ S ⇒ a - b, ab ∈ S. A subring…

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

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

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

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Q: A ring (R. +.) .) is commutative if addition is commutative in R. O True O False

A: Solve the following

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…

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Q: Let R be a finite commutative ring with identity. Then every prime ideal of R is maximal True O…

A: To prove that every prime ideal of R is maximal.

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

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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: Let R = {2n: n E Z} and define addition and multiplication O in R by a b = a + b and aOb = for all…

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

A: Given

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: 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…

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Q: Let CR,t,,) be a Commutative ring ?

A: First I recall you Definition of commutative ring.  A commutative ring is ring in which…

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: If N is an ideal of a ring R, which of the following statements is NOT always true? N is a subring…

A: Given : N is an ideal of a ring R

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

A: For a ring R, the element a∈R is idempotent if a2=a For a ring R, the element a∈R is nilpotent if…

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 that R is a commutative ring with unity 1. Then if ab is a zero divisor then a or b is a…

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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: (a) Let R be a commutative ring with M being maximal ideal in R then R/M is a field. (b) If R is a…

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Q: lisprove: that the triple <Z,,, constitutes a ring. et of units U(Z₂), and state whether (U(Z₁), Ⓒ)…

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Q: 6. Let R be a commutative ring with identity and let f be an onto homomorphism from R to a ring S.…

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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: Show if R is a commutative ring then R[x] is also a commutative ring.

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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 A, B,C be rings. Let & be a ring homomorphism from A into B and ß be a ring homomorphism from B…

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Q: Let A, B,C be rings. Let o be a ring homomorphism from A into B and B be a ring homomorphism from B…

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Q: Let A be a commutative ring with identity and D be an integral domain. Suppose that p: A → D is a…

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Q: 5- The image of a commutative ring under homomorphism is commutative a) True b) False a) True O b)…

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

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Q: Let R be a commutative ring with more than one element. Provethat if for every nonzero element a of…

A: We have given commutative ring R with more than one element. We have to prove R has a unity and…

Q: Suppose that R is a ring and that a2 = a for all a in R. Show that Ris commutative. [A ring in which…

A: Given: R is a ring such that a2=a for all a in R. To show: R is commutative ring.

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

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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…