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).