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
Existence and finiteness of a reduction number are guaranteed under mild conditions in Noetherian local rings;
Etymology and usage: in German-language texts the standard term is Reduktionszahl, while Reduktionstal is a nonstandard