. Which of the following pairs of ideals in Z[x] are comaximal? · (3), (5) •...
Solve problem 1 from Abstract Algebra dealing with ideals ,
prime ideals and maximal ideals in Ring theory.
Problem 1, Consider the ring 3 3 of integer pairs along with the prime ideal l # (3m, n) : m, n E ZJ. Prove that I is a maximal ideal of 3 x 3. 15 points Problem 2. Let R (R, be a commutativ ri
please be show detail and step by step but not over
complicate!
4) Describe all the ideals in the following rings (b) Q x Q (c) Z x Z.
4) Describe all the ideals in the following rings (b) Q x Q (c) Z x Z.
Every ring in this test is commutative with 1 and 1 0 1. Which of the followings are prime ideals of Z? (Separate your answers by commas.) A. ( B. (2). C. (9). D. (111). E. (101) 2. Which of the followings are ring homomorphisms? (Separate your answers by commas.) A.φ: Z → Z, defined by (n) =-n for all n E Z B. ф: Z[x] Z, defined by ф(p(z)) p(0) for all p(z) E Z[2] C. : C C....
Which of the following rings have complete factorizations? Select all that apply. • Z[V-5] Z[i] • Z[x, y, z) Which of the following rings have unqiue factorizations? Select all that apply. Z[V-5) Z[x, y, 2] • Z[x, y, 2]/(x2 - yz)
(3) Hint: Use the Euclidean Algorithm (repeated application of division algorithm using previous remain- ders) to find the greatest common divisor of the given pairs of elements and use that to express these principal ideals. (a) Express the ideals as 2178Z2808Z and 2178Zn 2808Z in Z as principal ideals. (b) Express the ideals (2r63 r+2x + 2) + (2r5 +3x4 + 4x +x+ 4) and (2r63z4 as principal ideals +2x +2)n (2r5 +34 + 4x3 + z2+4) in (Z/5Z)
(3)...
Consider the following point estimators, W, X, Y, and Z of μ: W = (x1 + x2)/2; X = (2x1 + x2)/3; Y = (x1 + 3x2)/4; and Z = (2x1 + 3x2)/5. Assuming that x1 and x2 have both been drawn independently from a population with mean μ and variance σ2 then which of the following is true... Which of the following point estimators is the most efficient? A. X B. W C. Y D. Z
3. In the following, consider z as a function of x and y, i.e., z = z(x, y) and use az az implicit differentiation to find the partial derivatives and ax ay (a) x2 + y2 + z2 = 3xyz (b) yz = ln(x + z)
Which of the following sets are finite? {x∈Z|x2≤10} {x∈Z|x3≤10} {x∈N|x3≤10} {x∈R|x2≤10} {x∈R|x3=10}
3. If the integers mi, i = 1,..., n, are relatively prime in pairs, and a1,..., an are arbitrary integers, show that there is an integer a such that a = ai mod mi for all i, and that any two such integers are congruent modulo mi ... mn. 4. If the integers mi, i = 1,..., n, are relatively prime in pairs and m = mi...mn, show that there is a ring isomorphism between Zm and the direct product...
Which of the following systems are inconsistent? (a) x + 2y + z = 2 (b) x + y + z 0 2x+2y-z=0 (c) -2x2 + 10 25 x2 +2x3 r2 (d) ax + by =