Checking the ring k[x,y,u,v]/(xy-uv) is a normal domain

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].

Exploring the Grothendieck ring K0 of endomorphisms

Abstract: The aim of this note is to compute the Grothendieck group K_0 of the category of endomorphisms. This computation mostly plays with linear algebra. The main result is that in K_0, every endomorphism f:P\to P is uniquely characterized by P and its characteristic polynomial \lambda_t(f). This computation was due to [1]. We will explain how to think about this computation, the reason for certain constructions and the “diagonalization” in this computation.

Edit history: fixed some mislabeling of diagrams (Mar 21 2022)

[1] Almkvist, G. (1974). The Grothendieck ring of the category of endomorphisms. Journal of Algebra28(3), 375-388.