Let R be a standard graded algebra over an infinite field \(\mathbb {K}\) and M a finitely generated \({\mathbb Z}\)-graded R-module. Let \(I_1,\ldots I_m\) be graded ideals of R. The functions \(r(M/I_1^{a_1}\ldots I_m^{a_m}M)\) and \(r(I_1^{a_1}\ldots I_m^{a_m}M)\) are investigated and their asymptotical behaviours are given. Here, \(r(\bullet )\) stands for the reduction number of a finitely generated graded R-module \(\bullet \).

中文翻译：

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关于理想产品的约数

令R为无限场\（\ mathbb {K} \）上的标准渐变代数，M为有限生成的\（{\ mathbb Z} \）渐变R-模。令\（I_1，\ ldots I_m \）为R的理想值。研究函数\（r（M / I_1 ^ {a_1} \ ldots I_m ^ {a_m} M）\）和\（r（I_1 ^ {a_1} \ ldots I_m ^ {a_m} M）\）及其渐近性给出了行为。在这里，\（r（\ bullet）\）表示有限生成的渐变R模块\（\ bullet \）的减少数。