The ideal for image of Segre embedding

Question

How to show the ideal $(X_{ij}X_{kl}-X_{il}X_{kj})_{0\le i,k\le m, 0\le j,l\le n}\subset k[X_{ij}]_{0\le i\le m, 0\le j\le n}$ is radical? I can show the zero locus defined by the ideal is the image of $\mathbf{P}_k^{m-1}\times\mathbf{P}^{n-1}_k\to\mathbf{P}_k^{mn-1}$, but I think this may not imply the ideal is radical. (The same question for the quadratic defining relations for Plücker embedding, why do they generate a radical ideal?)

Answer 1

$k[X_{ij}]_{0\le i\le m, 0\le j\le n}/(X_{ij}X_{kl}-X_{il}X_{kj})_{0\le i,k\le m, 0\le j,l\le n}$ is isomorphic to the Segre product of two polynomial rings in $m+1$, respectively $n+1$ indeterminates over $k$; see here. This is at its turn a subring of their tensor product, and therefore an integral domain.