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Saturday, January 9, 2016

UTM Ideals Varieties and Algorithm - Chapter 1 Section 4 Exercise 2

Problem:


Solution:

(i) $ \implies $ (ii)

If $ f_i \in I $, $ \forall i $, then $ \sum\limits_{i = 1}^{s}h_i f_i \in I $ because $ I $ is an ideal, therefore $ \langle f_1, \cdots f_s \rangle \subset I $.

(ii) $ \implies $ (i)

If $ \langle f_1, \cdots f_s \rangle \subset I $, then $ \sum\limits_{i = 1}^{s}h_i f_i \in \langle f_1, \cdots f_s \rangle \subset I $ because $ \langle f_1, \cdots f_s \rangle $ is an ideal. For $ k \in [1, s] $, set $ h_k = 1 $ and $ h_i = 0 $ for all $ i \ne k $, we get (i).

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