Does $\displaystyle <1>=<1+x,1+y>$ in $\displaystyle \mathbb{Q}[x,y]$?
I don't think this is true, how would I go about showing it?
Suppose it is $\displaystyle \Longrightarrow f(x,y)(1+x)+g(x,y)(1+y)=1$ , for some $\displaystyle f,g\in \mathbb{Q}[x,y]$ . Now apply the evaluation homomorphism $\displaystyle x\mapsto -1\,,\,\,y\mapsto -1$ to this equality and get a contradiction.