Show that the zero ideal is a product of maximal ideals (not necessarily distinct) in the ring $k[x,y,z]/(x(x-1),y^2,z^3)$.
I tried using Nullstellensatz and then the 4th Isomorphism Theorem, to get that the maximal ideals in this ring are of the form $(x-a_1,y-a_2,z-a_3)$ where $a_i\in k$. Then essentially I should get a product of these ideals to be equal $(x(x-1)\cdot p_1,y^2\cdot p_2,z^3\cdot p_3)$ where $p_i\in k[x,y,z]$, but I always end up with unwanted elements, e.g., $$(x(x-1),xy,xz,y(x-1),y^2,yz,z(x-1),yz,z^2)\not=(0).$$