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

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

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

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

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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: If R is a noncommutative ring with unity and x, y ∈ R, compute the product x(x + y)(x − y)y.

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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: 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: Let R, , O is a ring under two composion e and O üs follows ü e i; = a + b + 1 and aOb = ab + a + b…

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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 K is a simple ring, then K a Jacobsen radical ring. T F

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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: 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: If R is a commutative ring with unity and A is a proper ideal of R, show that R/A is a commutative…

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

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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: et R be a ring and assume a∈R is not a zero divisor. If ab=ac for some b,c∈R, then b=c

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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: If R is a commutative ring with unity, show that the characteristic of R[x] is the same as the…

A: If R is a commutative ring with unity, show that the characteristic of R[x] is the same as the…

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 R be a commutative ring with identity, and let a, b E R. Assume ab is a unit in R. Do a and b…

A: Here given R is a commutative ring with identity. and let a,b∈R and assume ab is a unit. we have to…

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

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

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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: 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: Let a, b, and c be elements of a commutative ring, and suppose thata is a unit. Prove that b divides…

A: Let a, b, and c be elements of a commutative ring where a is a unit. Suppose that bdivides c. Then c…

Q: Consider the ring R = [r, s,t) whose addition and multiplications tables are given below, Then ts S…

A: Given R=r,s,t be the ring under addition and multiplication. Given table is +rstrrstsstrttrs We have…

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

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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: 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.