7. a) Prove that every field is a principal ideal ring. b) Consider the set of numbers R {a+ bV2|a, bE Z}. Show that the ring (R,+, ) is not a field by exhibiting a nontrivial ideal of (R,+,).

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: Let R be a ring with unity in which r^2 = r for every r in R. Prover that if a does not equal 0 and…

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Q: Let R be a ring and define R* = ∆R = {(r, r) : r ∈R }.1 (a) Suppose R = Z6 and list the members of…

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Q: i)Prove that a field has only two ideals. ii)Prove that a commutative ring with unity that has only…

A: We will use the definition of ideal, field, and commutative ring to prove these results.

Q: 2. Let B and C be a ideals of a ring R. Prove that the set B - C = {teR such that t = f -g for some…

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Q: Let R be a ring. If the only ideals of R are {0} and R itself, then R is a field.

A: We know the definition of , ideals of ring R. A non empty subset I of R is said to be an ideal of R…

Q: Is it true or false that if I is a principal ideal in a commutative ring R with multiplicative…

A: Is it true or false that if  is a principal ideal in a commutative ring  with multiplicative…

Q: 5. Let R be a ring and I, J ideals of R. Is IUJ always an ideal of R? Either prove that this is the…

A: The objective of the question is to determine whether the union of two ideals I and J of a ring R is…

Q: Let M be an ideal of commutative ring R Prove that R/M is a field if and only if M is a maximal…

A: An ideal of a ring is a subset of the ring that is closed under addition and subtraction and that…

Q: 18. Let (R, ,) be a commutative ring with identity and let N denote the set of nilpotent clements of…

A: a). As given that N is nonempty as 0∈N.  an=0 (n= positive integer) Then r∈R ran=rnan=0 Let take a,…

Q: B. If R = Z[x] is the polynomial ring over Z, decide whether the following ideals are maximal ideals…

A: We shall use the following theorems to solve this problem.Theorem: Let R be a commutative ring with…

Q: 4) Let R be a ring and F-M₂(R) be ring of 2 x 2 matrices over R under standard addition and…

A: # Answer 4) Given:- R  is a ring.- F=M2​(R) , the ring of 2×2  matrices over R  under standard…

Q: Let T = { | EQ, a and b are relatively prime and 5 does not divide b}. Show that T is a ring under…

A: SOLUTION:  Let T=ab: ab∈ℚ,a and b are relatively prime and 5 does not divide b   We  have  show…

Q: Show that a commutative ring R with unity is a field if and only if its only two ideals are (0) and…

A: