quadric – Thinking Bear

Let $A= k[x,y,u,v]/(xy=uv)$ , is this ring normal (integral closed in its field of fraction)?

Edit (see comments below): Because $p$ is of height 1, then there exists one among $x,y,u,v,$ say $x$ such that $f(x)\notin p$ for any $f(x)\in k[x]$ . Otherwise, if there are polynomials in one variable for each of the variables $x,y,u,v,$ contained in $p$ , then $p$ is of height $>1$ . Thus $f(x)$ becomes invertible in $A_p$ for any $f(x)\in k[x]$ .