  A standard proof of Schur's Theorem yields that any $r$-coloring of
$\{1,2,\dots,R_r-1\}$ yields a monochromatic solution to $x+y=z$, where $R_r$
is the classical $r$-color Ramsey number, the minimum $N$ such that any
$r$-coloring of a complete graph on $N$ vertices yields a monochromatic
triangle. We explore generalizations and modifications of this result in higher
dimensional integer lattices, showing in particular that if $k\geq d+1$, then
any $r$-coloring of $\{1,2,\dots,R_r(k)^d-1\}^d$ yields a monochromatic
solution to $x_1+\cdots+x_{k-1}=x_k$ with $\{x_1,\dots,x_d\}$ linearly
independent, where $R_r(k)$ is the analogous Ramsey number in which triangles
are replaced by complete graphs on $k$ vertices. We also obtain computational
results and examples in the case $d=2$, $k=3$, and $r\in\{2,3,4\}$.
