A few days ago, I asked a linear algebra question, but it seems that the notions are better stated in terms of algebraic geometry. I don't have much solid knowledge of algebraic geometry, so I'm wondering if there is a basic explanation for the following.
Suppose you have homomorphism given by $$ \phi\colon\mathbb{C}[z_{11},\dots,z_{mn}]\to\mathbb{C}[x_1,\dots,x_m,y_1,\dots,y_n]: z_{ij}\mapsto x_iy_j. $$
Then is $\mathbb{C}[z_{11},\dots,z_{mn}]/\ker\phi$ integrally closed or not?
By integrally closed, I mean that $\mathbb{C}[z_{11},\dots,z_{mn}]/\ker\phi$ is equal to its integral closure (the set of elements of $k$ integral over $\mathbb{C}[z_{11},\dots,z_{mn}]/\ker\phi$) in its quotient field $k$.