Let $R=\mathbb{Q}[x_1,x_2,\ldots,x_n]$ be a ring of multivariate polynomials over $\mathbb{Q}$. Let $V=\mathop{\rm span}\{x_1,x_2,\ldots,x_n\}$ be a linear subspace spanned by the indeterminates. Let ${\frak p}\lhd R$ be a prime ideal of $R$ which trivially intersects $V$.

Question: Do we always have a maximal ideal ${\frak m}\lhd R$ such that ${\frak p\subseteq m}$ and ${\frak m}\cap V=\{0\}$?

Let $R=\mathbb{Q}[x_1,x_2,\ldots,x_n]$ be a ring of multivariate polynomials over $\mathbb{Q}$. Let $V=\mathop{\rm span}\{x_1,x_2,\ldots,x_n\}$ be the linear subspace spanned by the indeterminates. Let ${\frak p}\lhd R$ be a prime ideal of $R$ which trivially intersects $V$.

Question: Do we always have a maximal ideal ${\frak m}\lhd R$ such that ${\frak p\subseteq m}$ and ${\frak m}\cap V=\{0\}$?