Homework 4

Let $I$ and $J$ be ideals in $k[x_1,\ldots,x_n]$ and let $\prec$ be a term order on $k[x_1,\ldots,x_n]$.  

a) Define $IJ = \langle fg: f\in I, g\in J\rangle$.  Show that $\text{LT}_\prec(I)\text{LT}_\prec(J) \subseteq \text{LT}_\prec(IJ)$.

b) Let $G$ be a finite set of polynomials in $I$. Show that $G$ is a Gröbner basis for $I$ with respect to $\prec$ if and only if for all $f\neq 0$ in $I$ there exists $g\in G$ such that $\text{LT}_\prec(g)$ divides $\text{LT}_\prec(f)$.

c) (Bonus challenge!) Is the inclusion in part (a) ever strict?