The most general form of this theorem states that in a commutative unit ring , the height of every proper ideal generated by elements is at most . Equality is attained if these elements form a regular sequence.
Setting yields part of the original statement on principal ideals, also known under the German name Hauptidealsatz, that for every nonzero, noninvertible element of , the ideal of has height at most 1, and, moreover, iff is a non-zero divisor.
It immediately follows as a corollary that every proper ideal of a Noetherian ring has finite height and that a principal ideal domain has Krull dimension equal to 1.