Reduktionstal

Reduktionstal is a term used in commutative algebra to denote what is commonly called the reduction number of an ideal with respect to a reduction. Let R be a Noetherian ring and I an ideal of R. A subideal J ⊆ I is a reduction of I if there exists an integer n ≥ 0 such that I^{n+1} = J I^n. The smallest such n is denoted r_J(I), the reduction number of I with respect to J. The reduction number of I, written r(I), is the minimum of r_J(I) taken over all reductions J of I (when such reductions exist).

The concept measures how quickly the powers of I become governed by a smaller generating set. If I has a reduction generated by a minimal set of elements, the reduction number reflects the stabilization of the chain of powers I^n in terms of that reduced generating data. In practice, the reduction number is studied via related objects such as the Rees algebra and the associated graded ring gr_I(R), and it interacts with the Hilbert-Samuel function and multiplicity.

Existence and finiteness of a reduction number are guaranteed under mild conditions in Noetherian local rings; in particular, principal ideals have reduction number r(I) = 0 since I = J with J = I gives I^{n+1} = J I^n for all n, starting at n = 0.

Etymology and usage: in German-language texts the standard term is Reduktionszahl, while Reduktionstal is a nonstandard variant. See also Rees algebra, gr_I(R), Hilbert function, and multiplicity.