Let A be a Noetherian standard N-graded algebra over an Artinian
local ring A0β. Let I1β,β¦,Itβ be homogeneous ideals of A and M a
finitely generated N-graded A-module. We prove that there exist
two integers k and kβ² such that \mathrm{reg}(I_1^{n_1} \cdots I_t^{n_t}
M) \leq (n_1 + \cdots + n_t) k + k'
\quad\mbox{for all }~n_1,\ldots,n_t \in \mathbb{N}. Comment: 9 page