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Let $S_{\mathfrak{z}}(k,r)$ be the least positive integer such that for any$r$-coloring $\chi : \{1,2,\dots,S_{\mathfrak{z}}(k,r)\} \longrightarrow \{1,2, \dots, r\}$, there is a sequence $x_1, x_2, \dots, x_k$ such that$\sum_{i=1}^{k-1} x_i = x_k$, and $\sum_{i=1}^{k} \chi(x_i) \equiv 0 \pmod{r}$.We show that when $k$ is greater than $r$, $kr - r - 1 \leS_{\mathfrak{z}}(k,r) \le kr - 1$, and when $r$ is an odd prime,$S_{\mathfrak{z}}(k,r)$ is in fact equal to $kr - r$.

Upper and Lower Bounds on Zero-Sum Generalized Schur Numbers