Elements of ideals in the ring $K[x,y]$, where $K$ is a field

Question

Let $K$ be a field, and consider the polynomial ring $K[x,y]$. I want to show the followings:

1. $y \notin (x^2,xy,y^2)$

2. $x \notin (x^2,y)$

It seems that to show these, for example in 2, I should suppose $x \in (x^2,y)$ and derive a contradiction, but I have no idea to do this. Is there a straightforward argument to handle these kind of problems? Any hints will be appreciated.

Answer 1

For (2), every element of $(x^2, y)$ is of the form $x^2f + yg$. So set $x = x^2f+yg$. Note that $x \mid yg$, and since $x$ is prime, $x \mid g$. So we can write $g=xg'$. Now plug this into the original equation gives $x = x(xf+yg')$. Do you see what to do from here?

Answer 2

1. Every monomial in $(x^2, xy, y^2 ) $ is either divisible by $x^2$ or $xy$ or $y^2$. $y$ is not.

2. Every monomial in $(x^2, y ) $ is either divisible by $x^2$ or $y$. $x$ is not.