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Let $K$ be an algebraically closed field and $K[\underline{x}]$ its ring of polynomials in $n$ variables $x_1,\cdots, x_n$. Let $J\leq K[\underline{x}]$ be an ideal such that there are no monomials in $J$. Is there any characterization on a finite set of generators (probably a reduced Gröbner base) $G$ of $J$?

To rephrase my question, Is there a way to know when an ideal in $K[\underline{x}]$ has no monomials by just looking at a set of finitely many generators?

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    $\begingroup$ Let's think about the inverse: $J$ contains some monomial iff $J$ contains some power of $\langle \prod_{i=1}^n x_i \rangle$, iff $\sqrt{J}$ contains $\langle \prod_{i=1}^n x_i \rangle$, iff $V(J)$ is contained in $V(\langle \prod_{i=1}^n x_i \rangle) =$ the union of the coordinate hyperplanes. I don't suppose your $J$ is a prime ideal... $\endgroup$
    – Allen Knutson
    Commented May 11, 2011 at 3:39

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