Given a ring R with elements {0, 1, a, b}, where: a+a=b, a×a=b, and b+b=0. Find the result of (a+1)×(a+1).

Q: Let G=Z5xZ6X Z9 X Z12- Find the elementary divisors and invariant factors of G.

A: Solution :-

Q: (Q1)Find all idempotent elements and nilpotent elements of the following rings 2- (Z/(3),+..) + (C.…

A: We know that both given ring are field

Q: Find all possible pairs (a, b) in Z3 × Z3 such that Z3[r]/(r³+ar+b) is a field.

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Q: Show that the set Q(√2) = {a+b√2, a, b € Q} is a ring, under operations of multiplication and…

A: The set Q2=a+b2 , a , b∈Q  and operations under multiplication and addition define by :…

Q: Exercises :- 1- Show that: (Z24 / {0,6, 12 ,18},+,.) = (Z6, +6 r6) 2- Show that: (Z20 /{0,10},+,.)…

A: "Since you have posted a question with multiple sub-parts, we'll solve first 3 parts for you. To get…

Q: 4. Find a monic polynomial of least degree in ℝ[x] that has 1-i and 2i as roots. write the powers of…

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Q: Q1. a) Give an example of a semi-ring that is not a semi -algebra. b) Give an example of a…

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Q: Let a and b be arbitrary elements of a ring R, whose characteristic is 2 and ab = ba. Then (a) (a +…

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Q: CE Z3 A) Find all the values that make the Z[x]/(x² +x + c) quotient ring an object. B) Show that…

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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: Check if the solution is correct. Correct the mistakes. If x4=x for any x in R, prove that R is…

A: Given that x^4=x for any x in R. To prove R is commutative.

Q: Q2 Let (R, *0)be aring where (asb)* *(czd)=(a+c,b+d) and 2. (asb)o(e)d)= (acpbd, ad td) V…

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Q: Construct the multiplication table for the ring Z/6Z.

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Q: 3. Describe all units in the given ring: (c) Z/5Z (d) Z/6Z

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Q: Enter the multiplicative inverse of the element [21]49 X+ [18]49 in the ring Z49[x].

A: Extended Euclidean Algorithm

Q: Let a and b be arbitrary elements of a ring R, whose characteristic is 2 and ab = ba. Then (a) (a +…

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Q: Let R be a ring with unity. Assume b1a=1R and ab2=1R. Prove that b1=b2.   Also, let R be a ring…

A: Ring with unity,so 1R is different from 0R ,

Q: őf a ring R, e characteristic is 2 and ab = ba. Then コ+b)%=α'+b@ コーb?=α'+b? o!h (a) and (b) Ione of…

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Q: Q 10 Which one of the following is not a ring? (a) (Z, +, ·) (b) (M4x4(R), +, ) (c) (Zs,O5, 85) (d)…

A: We know that of G is group then G is follow closure property .  G is follow associative property.…

Q: Let a and b be elements of a ring R. Prove that the equation a+ x= b has a unique solution.

A: a and b are elements of ring R. We have to show that (a+x)=b has a unique solution.

Q: f = x² + x³x²+1, g = x³ + 1 in Q[x];

A: Since you have posted multiple questions, we will solve the first question for you. If you want any…

Q: Show that the centre of a ring R is a sub ring of R. And also show that the centre of a division…

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Q: Let R be a ring and a, be R. Show that ab is nilpotent implies that ba is nilpotent.

A: Solution. Since ab is nilpotent, (ab)n = 0 for some n ∈ N.

Q: a. Find the elements of Zg that have multiplicative inverses.

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Q: n the ring Z5 × Z7, determine −(3, 6) and (3, 6)−1 .

A: Here given ring is R = Z5 × Z7  We know that properties of ring R  If (a,b) is element of R then, i)…

Q: Write the multiplication table for the following commutative quotient ring: Z2[x]/(x³ + x² +x + 1).

A: Given that, ℤ2x/x3+x2+x+1 is a commutative quotient ring. We have to frame the multiplication table…

Q: Describe all ideals of the following quotient ring: Q[x]/(x² – 5x + 6)

A: Given:Q[x](x2-5x+6)