Editing Hilbert's basis theorem (section)

===Second proof=== 
Let <math>\mathfrak a \subseteq R[X]</math> be a left ideal. Let <math>\mathfrak b</math> be the set of leading coefficients of members of <math>\mathfrak a</math>. This is obviously a left ideal over <math>R</math>, and so is finitely generated by the leading coefficients of finitely many members of <math>\mathfrak a</math>; say <math>f_0, \ldots, f_{N-1}</math>. Let <math>d</math> be the maximum of the set <math>\{\deg(f_0),\ldots, \deg(f_{N-1})\}</math>, and let <math>\mathfrak b_k</math> be the set of leading coefficients of members of <math>\mathfrak a</math>, whose degree is <math>\le k</math>. As before, the <math>\mathfrak b_k</math> are left ideals over <math>R</math>, and so are finitely generated by the leading coefficients of finitely many members of <math>\mathfrak a</math>, say

:<math>f^{(k)}_{0}, \ldots, f^{(k)}_{N^{(k)}-1}</math>

with degrees <math>\le k</math>. Now let <math>\mathfrak a^*\subseteq R[X]</math> be the left ideal generated by:

:<math>\left\{f_{i},f^{(k)}_{j} \, : \ i<N,\, j<N^{(k)},\, k<d \right\}\!\!\;.</math>

We have <math>\mathfrak a^*\subseteq\mathfrak a</math> and claim also <math>\mathfrak a\subseteq\mathfrak a^*</math>. Suppose for the sake of contradiction this is not so. Then let <math>h\in \mathfrak a \setminus \mathfrak a^*</math> be of minimal degree, and denote its leading coefficient by <math>a</math>.

:'''Case 1:''' <math>\deg(h)\ge d</math>. Regardless of this condition, we have <math>a\in \mathfrak b</math>, so <math>a</math> is a left linear combination

::<math>a=\sum_j u_j a_j</math>

:of the coefficients of the <math>f_j</math>. Consider

::<math>h_0 =\sum_{j}u_{j}X^{\deg(h)-\deg(f_{j})}f_{j},</math>

:which has the same leading term as <math>h</math>; moreover <math>h_0 \in \mathfrak a^*</math> while <math>h\notin\mathfrak a^*</math>. Therefore <math>h - h_0 \in \mathfrak a\setminus\mathfrak a^*</math> and <math>\deg(h - h_0) < \deg(h)</math>, which contradicts minimality.

:'''Case 2:''' <math>\deg(h) = k < d</math>. Then <math>a\in\mathfrak b_k</math> so <math>a</math> is a left linear combination

::<math>a=\sum_j u_j a^{(k)}_j</math>

:of the leading coefficients of the <math>f^{(k)}_j</math>. Considering

::<math>h_0=\sum_j u_j X^{\deg(h)-\deg(f^{(k)}_{j})}f^{(k)}_{j},</math>

:we yield a similar contradiction as in Case 1.

Thus our claim holds, and <math>\mathfrak a = \mathfrak a^*</math> which is finitely generated.

Note that the only reason we had to split into two cases was to ensure that the powers of <math>X</math> multiplying the factors were non-negative in the constructions.