202110121137 Exercise 1.1.1

(a) Show that $x^2\in\langle x-y^2,xy\rangle$ in $k[x,y]$ ( $k$ any field).

(b) Show that $\langle x-y^2,xy,y^2\rangle = \langle x,y^2\rangle$ .

(c) Is $\langle x-y^2,xy\rangle = \langle x^2,xy\rangle$ ? Why or why not?

Definition.

Let $f_1,\dots ,f_s\in k[x_1,\dots ,x_n]$ . We let $\langle f_1,\dots ,f_s\rangle$ denote the collection $\langle f_1,\dots ,f_s\rangle = \{ p_1f_1+\cdots +p_sf_s:p_i\in k[x_1,\dots ,x_n]\}$ for $i= 1,\dots ,s$ .

(a)

$\begin{aligned} x^2 & = (x)(x-y^2) + (y)(xy) \\ & \in \langle x-y^2,xy\rangle \\ \end{aligned}$

(b)

First, we want to show

$\langle x-y^2,xy,y^2\rangle \in \langle x,y^2\rangle$ .

For any $p_1,p_2,p_3\in k[x,y]$ ,

$\begin{aligned} & \quad \langle x-y^2,xy,y^2\rangle \\ & = (p_1)(x-y^2)+(p_2)(xy)+(p_3)(y^2) \\ & = p_1x-p_1y^2+p_2xy+p_3y^2 \\ & = (p_1+p_2y)(x)+(p_3-p_1)(y^2) \\ & \in \langle x,y^2\rangle \\ \end{aligned}$

Next, we want to show

$\langle x,y^2\rangle\in\langle x-y^2,xy,y^2\rangle$ .

For any $p_1,p_2\in k[x,y]$ ,

$\begin{aligned} &\quad \langle x,y^2\rangle \\ & = (p_1)(x)+(p_2)(y^2) \\ & = (p_1)(x-y^2)+(0)(xy)+(p_1+p_2)(y^2) \\ & \in \langle x-y^2,xy,y^2\rangle \\ \end{aligned}$

All in all,

$\langle x-y^2,xy,y^2\rangle = \langle x,y^2\rangle$ .

(c) I guess $\textrm{\scriptsize{NOT}}$ .

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