I would like to confirm whether my interpretation of the definition of ideals generated by polynomials is correct, please.
In Ideals, Varieties, and Algorithms, Cox et al. define this as such (paraphrasing):
Given a polynomial ring over $n$ variables, $K[X = x_0,...,x_n]$, an ideal generated by a set of polynomials is:
$\langle f_1,...,f_m`\rangle = \bigl\{\sum^m_{i=1} h_i \cdot f_i \mid h_1,...,h_m \in K[X]\bigr\}$
So does this mean that I just multiply $f_1,...,f_m$ by $m$ arbitrary polynomials in $K[X]$, or do I multiply $f_1,...,f_m$ by ALL the elements in $K[X]$, as was specified in the following response:
DonAntonio's Response to "Understanding the ideal generated by a polynomial"
I think a better way of potentially phrasing my question is that it seems that the above definition just creates a large sum of the form:
$f_1h_1 + f_2h_2+ ...+f_mh_m$
So is an ideal generated by a set of polynomials the set of all sums of the above form, with different $h_1,...,h_m$? i.e:
$$ f_1h^{'}_1 + f_2h^{'}_2 + ... + f_mh^{'}_m\\ f_1h^{''}_1 + f_2h^{''}_2 + ... + f_mh^{''}_m\\ f_1h^{'''}_1 + f_2h^{'''}_2 + ... + f_mh^{'''}_m\\ \vdots $$
Thank you for your time!
You can imagine that the $f_i$ behave like a basis of a vector space. But here you do not use scalars for linear combinations, but arbitrary polynomials. This means for each element $f \in (f_1,\dots,f_m)$ we can find polynomials $h_1,\dots,h_m \in K[X]$ such that $f = h_1f_1 + \dots + h_mf_m$.