Polynomial-ring height and dimension formula ($\mathrm{ht}P+\dim (k[x_1,\dots ,x_n]/P)=n$)

For a prime ideal $P$ of a polynomial ring over a field, the quotient's Krull dimension is the transcendence degree of its fraction field, and its height is $n$ minus that degree. This dimension theorem identifies algebraic codimension with the length of prime chains. It must not be applied indiscriminately to every Noetherian ring without its polynomial-ring hypotheses.