Consider the monomial ideal $I=(vw,wx,xy,yz,zv)$ in the polynomial ring $k[v,w,x,y,z]$. How do we find the height of the ideal $I$ ?
Since $5=\dim k[v,w,x,y,z]=\mathrm{ht}(I)+ \dim (k[v,w,x,y,z]/I)$, so equivalently asking, how to find the Krull dimension of the ring $k[v,w,x,y,z]/I$ ?
Using the algorithms in this answer and this answer, we can compute a primary decomposition for $I$: $$ I = (v,x,y) \cap (v,x,z) \cap (w,y,z) \cap (v,w,y) \cap (w,x,z) \, . $$ Indeed, this shows that $I$ is an intersection of prime ideals (hence is radical). Each of these minimal primes has height $3$, so $I$ also has height $3$.
