The reduction theory goes back to the influential 1954 paper by Northcott and Rees, the paper that introduced the basic notions.
In algebraic geometry, the theory is among the essential tools to extract detailed information about the behaviors of blow-ups.
[1] For such ideals, immediately from the definition, the following hold: If R is a Noetherian ring, then J is a reduction of I if and only if the Rees algebra R[It] is finite over R[Jt].
A closely related notion is that of analytic spread.
, the minimum number of generators of J is at least the analytic spread of I.
[4] Also, a partial converse holds for infinite fields: if
This commutative algebra-related article is a stub.