1. Let I= {(x,y) | a, y € 2Z}. (a) Show that I is an ideal of Z × 2Z. (b) Use FIT for rings to show (Z x 2Z)/I = Z₂.
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- If you let R=Z[x] and I = (x^2 -2) be the principal ideal generated by f(x) = (x^2 -2). If r=(2x + I) exists in R/I, How do you prove that r^2=8 + I?If you let R=Z[x]and I = (x^2 -2) be the principal ideal generated by f(x) = (x^2 -2). If r=2x + I exists in R/I, How do you prove that r^2=8 + I?If R=Z[x] and f(x) = x2 + 1, remain true, but g(x) =x. How do you prove that in R/I, [g(x)] x [g(x)] = -1R/I I is still the principal ideal generated by f(x)
- if you let r=z[x] and i = (x^2 -2) be the principal ideal generated by f(x) = (x^2 -2). How would you find an element s ϵ R/I, such that s^2 = 8 + I.Let I ⊆ R be an ideal.(a) Prove that every element of R/I is a solution of x2 = x if and only if r2 − r ∈ I for all r ∈ R.Is R/I an integral domain?(b) Suppose that R is an integral domain. Is R/I necessarily an integral domain? If so, prove it.If not, provide a counterexample.If R = Z3 x Z4 and a = (2,2), which is in R.How do you find the elements of the principal ideal generated by a? What are the elements of the coset (1,3) + i? How many elements are in the quotient ring?
- (a) express ux, u y, and uz as func-tions of x, y, and z both by using the Chain Rule and by expressing u directly in terms of x, y, and z before differentiating. Then (b) evaluate ux, u y, and uz at the given point (x, y, z). u = e^(qr) sin-1 p, p = sin x, q = z^2 ln y, r = 1/z; (x, y, z) = (pai/4, 1/2, -1/2)What is the unity in R/I where R=ℤ_10 and I={0,2,4,6,8} is an ideal in R?this question is differential geometry Let F: IR3 →IR3 is a diffeomorphism and M is a surface in IR3 , prove that the image F(M) is also a surface in IR3.
- Let φ1 = 1, φ2 = x, φ3 = x2 on the interval 0 ≤x ≤1. Find the following inner products:(a) < φ1,φ2 >(b) < φ2,φ3 >(c) Distance between φ1 and φ2Let I := {2a + xf(x)|a ∈ Z, f(x) ∈ Z[x]} ⊆ Z[x]. Show that (i) I is an ideal of Z[x] and (ii) it is not principal.In Z{x}, Let I = {f(x) ∈ Z[x] / f(0) is an even integer} Prove that I = <x,2> is I prime ideal of Z{x}? is I a maximal ideal of Z{x}? How many elements does Z{x}/I have?