Say $P$ is an ideal of the ring $k[x_1,...,x_n]$ where $k$ is any field. Moreover say $G$ is a reduced Gröbner basis of $P$ (for some monomial ordering). Is it true that $P$ is prime if and only if $f$ is irreducible for any $f\in G$? If it is false, does the claim hold for some monomial ordering (like an elimination order)?
What about if $k$ is algebraically closed?
Here's a counterexample for the case where $k$ is not algebraically closed . . .
In the ring $\mathbb{R}[x,y]$, consider the ideal $$I = (x^2 + 1,y^2 + 1)$$ The ideal $I$ is not a prime ideal, but is already Groebner reduced for any monomial ordering.