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.
