# Math Help - Generated Ideals

1. ## Generated Ideals

Does $<1>=<1+x,1+y>$ in $\mathbb{Q}[x,y]$?

I don't think this is true, how would I go about showing it?

2. Originally Posted by skamoni
Does $<1>=<1+x,1+y>$ in $\mathbb{Q}[x,y]$?

I don't think this is true, how would I go about showing it?

Suppose it is $\Longrightarrow f(x,y)(1+x)+g(x,y)(1+y)=1$ , for some $f,g\in \mathbb{Q}[x,y]$ . Now apply the evaluation homomorphism $x\mapsto -1\,,\,\,y\mapsto -1$ to this equality and get a contradiction.

Tonio