Factoring a homogeneous element in graded ring

Question

Let $k$ be a field, and $A = k[w,x,y,x] / (wz-xy)$, which is an integral domain. I would like to show that if $h$ is a homogeneous element in $A$, not irreducible, then it factors into a product of non-trivial homogeneous elements. Could someone please show me how I can do this? Thanks!

Answer 1

Hint. Suppose $h=uv$ is a factorization with $u$ and $v$ not necessarily homogeneous. Write $u$ and $v$ as sum of nonzero homogeneous elements, distribute the product $uv$ and look at the term of maximal degree there.