## Abstract

We extend the regularity criterion of Bayer-Stillman for a graded ideal $\mathfrak {a}$ of a polynomial ring $K[\underline {\bf x}] := K [\underline {\bf x}_0, \dots , {\bf x}_r]$ over an infinite field $K$ to the situation of a graded submodule $M$ of a finitely generated graded module $U$ over a Noetherian homogeneous ring $R = \oplus_{n \geq 0}R_n$, whose base ring $R_0$ has infinite residue fields. If $R_0$ is Artinian, we construct a polynomial $\widetilde{P} \in {\mathbb Q}[{\bf x}]$, depending only on the Hilbert polynomial of $U$, such that $\operatorname{reg}(M) \leq \widetilde{P} ( \max \{ d(M), \operatorname{reg}(U) + 1 \} ) $, where $d(M)$ is the generating degree of $M$. This extends the regularity bound of Bayer-Mumford for a graded ideal $\mathfrak {a} \subseteq K[\underline {\bf x}]$ over a field $K$ to the pair $M \subseteq U$.