(4) The ring (Z, +,.) has the following not maximal ideal (a) ((11), +,.) (b) ((31), +,.) (c) ((0), +,.) (d) No Choice O (a) O (b) O (c) O (d)

Q: Suppose L: R² R³ is defined by L the given bases. [L(u)]s, = [L(u)]B₂ = ~Q[*^~-{]}]}----(HH:)…

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Q: (c) 2, 3, 4+ V10 and 4. V10 are irreducible elements of R.

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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: 7. If f(x) is irreducible over R, then f (x²) is irreducible over R. a) True b) False

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Q: (8) In the ring (Z, +o6), we get n {(a): a prime element in Z,}: (a) o (b) ((0), +66) (c) (Z6, +66)…

A: Prime Ideal: Let R, +, · be a commutative ring. An ideal A of R is said to be a prime ideal of R if…

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: One of the following is not integral domain a) (Z, +7,7) b) (R, +,) c) (C,+,) d) (P(X),A,0) a) O d)…

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

Q: Problem #5: Suppose that an n x n matrix A has a QR-decomposition where Problem #5: R = 2 0 0 -2 1 0…

A:

Q: Show that if '0' = '1' in a Ring (R, +, *) then R has only a single element. Here, '0' is the…

A: Consider 0 is the additive identity for a ring, for any element  x in the ring. Then…

Q: The number of reducible monic polynomials of degree 2 over Zz is

A: The number of reducible monic polynomials of degree 2 over Z3 is

Q: Q2/(A): Let L.N are submodules of M and n:M - M/N be the natural R-homomophism. Is n(L) submodule of…

A: Given that,                 L and N are submodules of M .  Also, π : M → MN is a natural R-…

Q: Let R be a ring and consider RxZ={(r,n)|r e R, n e Z}. Define (r,n) + (s,m) = (rts, n+m) (r,n)(s,m)…

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Q: Let f(x) = x' + 2x + 4and g(x)= 3x + 2 in Z,[»). Then, upon dividing f(x) by g(x), we get 7- The…

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Q: (8) In the ring (Z,, +6,•6 ), we get n {(a): a prime element in Zg} ...... (а) ф (b) ((0), +6, ·6 )…

A:

Q: For the following monic polynomial P(z) = zn+an−1z"−¹+...+a₁z+ao, show that all the roots lie in the…

A: Thus, all the zeros of P(z) whose modulus is greater than 1 lie in |z|≤(1+∑n−1k=0|ak|2)1/2.Those…

Q: Theorem: if I is a Positive implicative Bck-algebra Then (Lx),,Lo) is a BCK-algebra

A: Consider the given theorem, '' if X is positive implicative Bck-algebra then Lx,⊙,L0 is a…

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: 4. (a) Solve the set of congruences х 1(mod 3), x= 2(mod 4), x = 3(mod 5).

A: We  will solve it by Chinese Remainder Theorem Given  x≡1(mod3)x≡2(mod4)x≡3(mod5) Compare it with…

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: Q1) For f(x), g(x), and Z„[x] given in Exercises 7-10, find the greatest common divisor d(x) of f(x)…

A: Given - f(x)=x3+x2+2x+2 and g(x)=x4+2x2+x+1  We have to find greatest common divisor d(x) for the…

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

A: Field of quotient

Q: 5. In the ring (Z, +66) we have Z6/(3) forms... ..... ..... (a) ring (b) integral domain (c) field…

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Q: În a ring R, show that i) Oa = a0 = 0 ii) a(- b) = (- a)b=- (ab).

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

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Q: - If A is an ideal of R and x, y e R then prove that (i) x e A A+x=A (ii) A +x = A +y e x-ye A.

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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…

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Q: 4. Let X1, ,X4 be iid Unif(0, 0). (a) Show that X(4) is independent of X(1)/X(4). (b) Show that X(4)…

A: Given that X1,X2.............,X4 be iid Unif0,θ (a) X4 is complete sufficient statistics and…