All rings are commutative, and k is an algebraically closed field.1. Let R = k[x1,...,x,). Verify the following statements.(a) Let {Ia} be a collectionideals in R. Show that Na V(Ia) = V(Uala).(b) Let I, J CR be ideals. Show that V(IJ) = V(I) UV(J).2. Let J C k[x1,...,xn] be an ideal and X and Y algebraic sets. Verify the followingstatements.(a) V(I(V(J))) = V(J).(b) I(V(I(X))) = I(X).(c) X = Y if and only if I(X) = I(Y).

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Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.4: Ordered Integral Domains

Problem 1E: Complete the proof of Theorem 5.30 by providing the following statements, where and are arbitrary...

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All rings are commutative, and k is an algebraically closed field. 1. Let R= k(x1,...,xn]. Verify the following statements. (a) Let {Ia} be a collection of ideals in R. Show that . V(Ia) = V(UaIa). (b) Let I, JC R be ideals. Show that V(IJ) = V(I)UV(J). %3D