Ideals modulo a prime

The main focus of this paper is on the problem of relating an ideal $I$ in the polynomial ring $mathbb Q[x_1, dots, x_n]$ to a corresponding ideal in $mathbb F_p[x_1,dots, x_n]$ where $p$ is a prime number; in other words, the textit{reduction modulo $p$} of $I$. We first define a new notion of $sigma$-good prime for $I$ which does depends on the term ordering $sigma$, but not on the given generators of $I$. We relate our notion of $sigma$-good primes to some other similar notions already in the literature. Then we introduce and describe a new invariant called the universal denominator which frees our definition of reduction modulo~$p$ from the term ordering, thus letting us show that all but finitely many primes are good for $I$. One characteristic of our approach is that it enables us to easily detect some bad primes, a distinct advantage when using modular methods.