Let $W$ be a finite-dimensional ${\bb Z}/p$ -module over a field, ${\bf k}$ , of characteristic $p$ . The maximum degree of an indecomposable element of the algebra of invariants, ${\bf k}[W]^{{\bb Z}/p}$ , is called the Noether number of the representation, and is denoted by $\beta(W)$ . A lower bound for $\beta(W)$ is derived, and it is shown that if $U$ is a ${\bb Z}/p$ submodule of $W$ , then $\beta(U)\le \beta(W)$ . A set of generators, in fact a SAGBI basis, is constructed for ${\bf k}[V_2\oplus V_3]^{{\bb Z}/p}$ , where $V_n$ is the indecomposable ${\bb Z}/p$ -module of dimension $n$ .