Let $k[x_1,\ldots,x_n]$ be a polynomial ring, $k$ be an algebraically closed field. Suppose $k[T_1,\ldots,T_m]$ is a finitely generated $k$-subalgebra such that for any proper ideal $I$ of $k[T_1,\ldots,T_m]$, $Ik[x_1,\ldots,x_n]\neq(1)$ .
I want a counter example such that the above is not true. Namely, find a finitely generated $k$-subalgebra $k[T_1,\ldots,T_m]$ of $k[x_1,\ldots,x_n]$ such that there exists a proper ideal $I$ of $k[T_1,\ldots,T_m]$, but the extended ideal $Ik[x_1,\ldots,x_n]=k[x_1,\ldots,x_n]$.
I believe there exists many examples, but I have no one in hand now.
Motivation: Consider $\varphi:\mathbb{R}\to \mathbb{R}$, $x\mapsto x^2$. Then it is clear that $\mathrm{\varphi}$ is not an algebraic set. So I assume $k$ be an algebraically field, and want to know if an image of a polynomial map is an algebraic set.
Let $k$ be an algebraically closed field. Suppose $\varphi:k^n\to k^m$ is a polynomial map defined by $(T_1,\ldots,T_m)$, I want to know if the $\mathrm{Im}(\varphi)$ is always an algebraic set. The polynomial map $\varphi$ induces a ring homomorphism $\widetilde{\varphi}$ from $k[x_1,\ldots,x_m]$ to $k[x_1,\ldots,x_n]$. Let $\mathfrak{p}=\mathrm{Ker}\widetilde{\varphi}$, then $Z(\mathfrak{p})=\mathrm{Im}(\varphi)^-$(the closed closure of $\mathrm{Im}\varphi$). We know that {points in $Z(\mathfrak{p})$} 1:1 correspond {maximal ideals of $k[x_1,\ldots,x_m]/\mathfrak{p}$} 1:1 correspond {maximal ideals $(T_1-b_1,\ldots, T_m-b_m)$ of $k[T_1,\ldots,T_m]$}, then we know $Z(\mathfrak{p})=\mathrm{Im}\varphi$ iff for every maximal ideal $\mathfrak{m}$ of $k[T_1,\ldots,T_m]$, $\mathfrak{m}^e\neq(1)$ in $k[x_1,\ldots,x_n]$.
So I ask the above question.
When I considered one indeterminate polynomial ring, I found that there exist no counter examples. As a consequence, I got an interesting question for linear algebra as following.
Let $f,g\in \mathbb{Q}[x]$, suppose $(f,g)=1$, then there will exist $h(u,v)\in \mathbb{Q}[u,v]$ and $l(u,v)\in \mathbb{Q}[u,v]$ such that $f(x)h(f,g)+g(x)l(f,g)=1$. I will ask how to find $h,l$ ?
Thanks.