Question 3. > Given the triple < Zx 3Z, +,- defined in terms of the Cartesian product Z x 3Z of two rings Z, and 3Z under operations (k, 3m) + (1,3n) = (k+1,3m +3n) and (k, 3m) - (1,3n) = (kl. 9mn) for k, l, m, n € Z. (a) Prove that < Z x 3Z, +, -> is a ring. Is it a commutative ring? Justify your answer. (b) Does the triple < Zx 3Z, +, > form an integral domain? Justify your answer.

Q: Let (x, y) stands for a complex number z E C where x = Re(z) and y = Im(z). Now, let us define a new…

A:

Q: Explain why the operation (A, B)- tr(AB) defines an inner product on S, but not on Mn.

A: We know that M22 is the vector space of all 2×2 matrices with real entries and S2 is the space of…

Q: (5) The ring (Z6, +6,•6 ) has that rad Z, (a) ((0), +6.-6 )

A: Given: To explain the correct option of the given statement as follows,

Q: Determine: 1 [1 -1 2]}-1 [2 3]+|0 1 -1

A: Step:-1 Given that  1-124-1023+1001-1-1Take U=1-124-1023+1001-1-1Since, we…

Q: Write the conjugate of va + 3

A: Definition of conjugate.

Q: 2. Let (X,(x) and (Y, (,)) be Hilbert spaces. For (x1,y1), (2, y2) = X x Y, put ((x1,y1), (2, 2))xxy…

A:

Q: Let (R, +,-) be a ring and a, b E R. Prove the following. *)If ab = -ba, then (a + b)² = a² + b².…

A: Ring and integral domain

Q: Find the values of w,x,y and z in fraction form using a)gauss elimination, b) gauss-jordan…

A: Since you have posted a multiple question according to guildlines I will solve first question for…

Q: Why is inversion [that is, (x, y) = (-x, -y)] not listed as one of the four kinds of isometries in…

A: We have to tell, Why is inversion [that is, ϕx,y=-x, -y] not listed as one of the four kinds of…

Q: Give an example of each of the following, and show that your example satisfies the "highlighted"…

A: For the solution follow the next steps.

Q: Z3-22 If z1 = -4 - j5, Z2 = -3 + j2 , and z3 = 2 - j3, what is Re Note: z* is the complex conjugate…

A:

Q: True or False. The set {(0,0),(1,0),(2,0),(3,0),(4,0)} is a principal ideal of z5 x z3

A: Ideal:  A non - empty set S of ring R  is said to be an ideal of ring R if, (i)    a - b  ∈ S ;  ∀…

Q: One of the following is an ideal of z1: (0,3} (0,2,4,6} None (9,3,6,9}

A: From given statement

Q: 1. The product of 3x²_5x² + 3 and 2x + 5x – 4 in z,[x]/ is а) 2x b) 2x + 1 c) 2x + 2 d) 2x + 3

A: .

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

A:

Q: ments is defined as N(a+bV2) = |a^2 ^2].Prove that for every a, bER (b 6= 0)it is sible to find q,…

A: Sol

Q: True or False: (p, q) = p(1)q(1)+p(2)q(2) +p(3)q(3) defines an inner product on P2(R).

A:

Q: For what values of a does the homomorphism system (b – 1)x + 2y = 0 2x + (b – 1)y = 0 Have a…

A:

Q: 5. In the ring (Mat,x2(Z), +,.), if A € Matzx2(Z) then the ideal (A) = (a) {B. A: BE Matzx2(Z)} (b)…

A:

Q: Let T={[a, ) : a€R}, then T is not Borel algebra.

A:

Q: 4. The field of quotients of (Z2, +2)-2) is (a) Q2 (b) Z2 (c) Z4 (d) Zg

A: Field of quotient

Q: (4) The ring (Z, +,.) has the following not maximal ideal (a) ((11), +,.) (b) ((31), +,.) (c) ((0),…

A: Let R be a ring. A two-sided ideal I of R is called maximal if I ≠ R and no proper ideal of R…

Q: Show that for any k-algebras A1, A2,..., A,, we have (A1 x A2 x ...x A, )P S A" x A" x as algebras.

A: Let, A1,+,*,A2,+,*,...,Ar,+,* be the K-algebras, where + is the addition from 1 to r o which A1 is…

Q: + Bz³) ûx + y)ûz

A:

Q: QUESTION 10 Consider a ring 7I/2]={a + b[/2]| a,bEZ} where addition and multiplication on ZI/2jis…

A:

Q: (8) In the ring (Z,, +6, •6 ), we get n{(a): a prime element in Z6} = ...... (a) $ (b) ((0), +6, 6 )…

A:

Q: a b r s - Let R а, b, d 0 d r, s, tE Z, s and S even}.ir If S is an ideal of R, what can you say…

A:

Q: 7. Let V={(x,y) x,y € R}. Suppose addition and scalar multiplication are defined using the following…

A: