(3) Let A be commutative ring with identity, then A has just trivial ideals iff A is ....... O integral domain

Q: IF J is nil left ideal in an Artinian ring R, then J is

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Q: Let x, y, and z be an element of a ring Prove that x(y-z) = xy-xz and (y-z)x=yx-zx

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Q: 9. Let R be the ring Q[x]/(x³ - x² − x + 1). (a) List all the ideals of R and draw a diagram showing…

A: It is given that R=Qxx3-x2-x+1. (a) The objective is to list all ideals of R and draw a diagram that…

Q: 7. Let R be a commutative ring and let F = F(R, R) be the set of functions f: R→ R. Functions in F…

A: let R be a commutative ring and F=FR,R be the set of function f:R→R defined as f+gx=fx+gxfgx=fxgx…

Q: 7. Suppose that (R, +..) be a commutative ring with identity and (I, +,.) be an ideal of R. If I is…

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Q: Give an example of ring elements a and b with the properties thatab = 0 but ba ≠ 0.

A: We have to give the example of ring elements that follow the following properties: ab=0 but ba≠0.

Q: (6) Let R be a commutative ring with a one, and let J be an ideal of R. Show that if R/J is a field,…

A: Let R be a commutative ring with a one, and let J be an ideal of R.  We have to show that if RJ is a…

Q: Which of the following are ring (i): f: (2Z, +,-) → (Z, +,-), (ii): ƒ: (Z8, +,-) → (Z4, +,-), - 2x -…

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Q: Let x, y, and z be an element of a Ring Prove that (y-z)x = yx - zx

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Q: Show that in the factor ring Z[i] / , the element x + is a unit.

A: Given a factor ring Zx2x+1=a0+a1x,+⋯+2x+1 |a1∈Z  As 2x+1+2x+1=0+2x+1 Since,  2x+1∈2x+1   as aH=H iff…

Q: 5) Suppose that (R, +,.) be a ring without identity and has a subring with identity, then (a) R…

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Q: Is the ring Z/6Z and Z6 are isomorphic.

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Q: True or False: Any ring must be commutative with identity.

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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: 1. Consider the proof of the backwards direction of the theorem below. Justify each step that makes…

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Q: Let (R, +, .) be a nontrivial ring with * identity, prove that 1 0

A: It is given that (R,+, .) be a nontrivial ring with identity. Now we have to show that 1≠0. So, (R,…

Q: M„(Z) D This Option

A: Clearly the ring Mn(F) is infinite iff F is an infinite field.

Q: Is the idcal (x² + 1, x + 3) C Z[x] a principal idcal? Explain. The ring Z[x]/(x² +1, x+3) is…

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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: Prove that the quotient ring is a finite field.

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Q: a Show that C = b a commutative ring with unity even if M₂ (R) is not. Finally, show that C is a…

A: Consider the ring M2ℝ=abcd/a, b, c, d∈ℝ. We have given that C=a-bba/a, b∈ℝ. We need to show that C…

Q: (1) Let a be a fixed element in the commutative ring R, and let A- (xe R: xa = 0). Prove that A is…

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Q: The ring Z is isomorphic to the ring 3Z True False

A: The ring Z has identity 1 as 1·a=a·1=a∀a∈Z The ring 3Z has no identity i.e. there does not exist…

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: Suppose R is a commutative ring with identity 1R. Show that R is a field if and only if its only…

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