Let $A$ be a domain and let $I$ be the ideal in $B=A[x_1,\dots,x_n,y_1,\dots,y_n]$ generated by all the $2 \times 2$ minors of the matrix $$\begin{pmatrix} x_1 & x_2 & \dots & x_n\\ y_1 & y_2 & \dots & y_n \end{pmatrix},$$ i.e. $I = (x_iy_j-x_jy_i : 1\le i<j\le n)$. I tried to show that $I$ is prime by showing that $B/I$ is a domain. Unfortunately, I have no idea how to prove this. Maybe someone can give me a hint?
This is problem 7.7 in the textbook Binomial Ideals by Herzog, Hibi, Ohsugi.
It’s much easier to answer this question if you know some geometry (apologies if you want an answer using strictly algebra, but I think it’s more instructive).
View the ideal $ I $ as a homogeneous ideal of $ R[ y_1, \ldots, y_n] $ where $ R $ is $ \mathbb{C} [x_1, \ldots, x_n] $. Then the equations define the blow-up of $ \mathbb{A}^n $ at the origin, as a closed subscheme of $ \mathbb{P}^{n-1} \times \mathbb{A}^n $. This blow-up is also the total space of the tautological line bundle of $ \mathbb{P}^{n-1} $ and hence, irreducible and reduced, that is, a variety itself. So the ideal $ I $ is a homogenous prime ideal, hence is a prime ideal if you forget the grading.