5. In the ring (Mat,x2(Z), +,.), if A € Matzx2(Z) then the ideal (A) = (a) {B. A: BE Matzx2(Z)} (b) {B. A + n. A: BE Matzx2 (Z) and n e Z} (c) Matzx2(Z) (d) No Choice (a) (b) (c) O (d) 6. In the ring (Z99, +90,-90 ), the ideal (10) n (15) = (a) (80) (b) (75) (c) (60) (d) No Choice

Q: 2. Using OLS or any other suitable method, Prove that E(ß1) = B1 E(ßo) = Bo

A:

Q: Show that MnXn and Rn^2 are isomorphic.

A:

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: (3) The ring (4Z, +,.) has the following prime ideal ... (a) ((0), +,.) (b) ((8), +,.) (c) ((12),…

A:

Q: 4. Prove that for n > 0: E, 2* = 2"+1 – 1. sk3D0

A:

Q: 5.Use thelefinition to Rrave that Jw wh- 2

A:

Q: 2. Find all volves of tand u x(t,4)=0 and simultomebusly such that =0. 3. 5x-11* C.

A: Get the equations for:  fx=0fy=0 15x2-11y=0-11x+3y2=0 or 15x2=11y3y2=11x

Q: 5. Prove OR disprove that Z/2ZOZ/3Z is isomorphic to Z/6Z

A: To prove or disprove that  Recall that , Now to prove or disprove that  These elements of these…

Q: shor w that To- {0} abelian 7oup. where p an is prime.

A:

Q: Prove that if |z| = 3, then %3D 5 2z – 1 7 - 13 4+ 22

A: The given is related to complex numbers. The solution is explained below.

Q: b)Show that, 6(z, 3zy – y' and (z,y) = -+3ry are harmonie

A:

Q: (7) In the ring (Z, +, .), we get n {P:P non trival prime ideal in Z}: (a) o (b) (Z, +, .) (c)…

A:

Q: (7) In the ring (Z, +,.), we get n {P: P non trival prime ideal in Z} %3D ....... (a) o (b) (Z, +,.)…

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Q: This exercise provides an alternative to Theorem 8.2 for factoring an r-cycle into transpositions.…

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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: 2) A ving having no proper ideals Called single Ying is D True O Fulse i

A: A ring having no proper ideal is called single ring. to determine whether the statement is true or…

Q: Show that |1 - (x, + x,) - (x, + x,) X,*z = (x, – ,) (x, – y2) (x2- ,)(x2 - 2) 1 -(), + x2) 1 - (y,+…

A: See the attachment

Q: 3. How many units are in the ring (Z/15Z) × (Z/125Z)?

A:

Q: 1. Suppose that (R x R, +,.) is a field where (x, y) + (z, w) = (x + z, y + w) and (x, y). (z, w) =…

A: Suppose that ℝ×ℝ, +, · is a field where (x, y) + (z, w) = (x+z, y+w) and x, y·z, w = xz-yw, xw+yz…

Q: (2) Suppose that G, C Zn-r. Then prove that G,-1C Zn-r+1

A: We prove that

Q: {0, A, Aº, $2} and X2 = {¢, B, Bº, S2} are o-fields, but &¡ U X2 is not. Why?

A:

Q: verify that (AT )T = A

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Q: in(Zg++s)indeyclic grauf2enerated by I, Z,3

A:

Q: Prove that F(z) = 81 8z + 1 is chaotic on (-1,1).

A: To prove: Fx=8x4-8x2+1 is chaotic on -1,1. First plot graph of given function with -1,1 as follows:

Q: Suppose X has n elements and Y has m elements. How many bijections X → Y are there?

A: To find: The number of bijections X → Y are there. Given information: X has n elements and Y has m…

Q: 2. In the ring (4Z, +,.), the ideal (8) is (a) not prime (b) maximal (c) maximal and not prime (d)…

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Q: Verify Eular,s theorem for z =ax2+2hxy+by2

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Q: such that ab = 1 b) : In Z3XU3, Find the order of (1, 2). Show your work.

A:

Q: Q2: Show that whether (Z16,O.O) is field or not?

A: Properties of a field For all a,b, and c∈F, the following hold: a+(b+c)=(a+b)+cand a⋅(b⋅c)=(a⋅b)⋅c…

Q: 4. Prove C(n, r) = C(n- 1,r- 1) +C(n- 1, r) %3D using a combinatorial proof.

A: We know, C(n,r) = n!/((n-r)!r!)

Q: _. . Prove that z² = + Emo(-1)" (n + 1) (²2)", where |z-21 < 2.

A:

Q: c If in is an involution of X, show that (x, y)in = (yin, xin) is an involution of X2). %3D

A: Let X be a lattice We have to show that (x, y)in=(yin, xin) is an X[2] X[2] is a 2 x 2 matrix We…

Q: Fiven with z1 = (-3,1) and z2 = (1, 4). Show that |21 2| = |21||2|- %3D

A: Complex number and it's modulus

Q: not a Field Always because if we take the ideal I = : Z₂ —————→ Z₂ such that y(x) = { 1 if x is odd…

A:

Q: Explain why 3Z is not an ideal in Q

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Q: 4Z is prime ideal of 2Z, but it is not Maximal ideal of Z. To fo

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Q: Verify Eular's Theorem for z=ax²+2hxy+by2

A: According to Euler theorem,  x∂z∂x+y∂z∂y=nz Where n is the degree of the homogeneous equation.

Q: 2- (a) Using [X, P] = ih, show that [X, PJ = 2ih& and [&, Þ³1= 2ihP. (b) Show that [, p2 = 2in(in +…

A:

Q: (3) The ring (4Z, +,.) has the following prime ideal.. (a) ((0), +, .) (b) ((8), +,.) (c) ((12),…

A:

Q: 1. Suppose that (R x R, +,.) is a field where (x, y) + (z, w) = (x +z, y + w) and (x, y). (z, w) =…

A:

Q: How can I find all monic divisors of degree 1 in Z_7 [x], 5x^2+5x+4? please help me get started with…

A: Set of all Polynomials over a field ℤ7 is given by ℤ7x=anxn+an-1xn-1+ . . . +a1x+a0  where a0,a1, .…

Q: 2. Suppose that {v1,... , Vp} is an orthonormal set in R", and that x E R". What can you say about…

A:

Q: 1. Suppose that (Q x Q, +,.) is a field where (x, y) + (z, w) = (x + z,y + w) and (x, y). (z, w) =…

A:

Q: If both z + w and zw are real, how to prove that either z = w (conjugate) or Im(z) = Im(w) = 0?

A:

Q: 2. If A + 4, N, what is the Borel field generated by {A}.

A: Hello. Since your question has multiple parts, we will solve first question for you. If you want…

Q: What is the principal ideal (3) of z8 = {0,1,2,3,4,5,6,7}? Is it not just z8?

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Q: In Z[V-5], the element 29 is: None of the choices Irreducible Unit reducible

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Q: 1. Suppose that (R x R, +,.) is a field where (x, y) + (z, w) = (x + z, y + w) and (x, y). (z, w) =…

A: To find the inverse element of (1, 2) first we find the identity element in the given field, and…