  Let $I \subseteq R = \mathbb{K}[x_1,\ldots,x_n]$ be a toric ideal, i.e., a
binomial prime ideal. We investigate when the ideal $I$ can be "split" into the
sum of two smaller toric ideals. For a general toric ideal $I$, we give a
sufficient condition for this splitting in terms of the integer matrix that
defines $I$. When $I = I_G$ is the toric ideal of a finite simple graph $G$, we
give additional splittings of $I_G$ related to subgraphs of $G$. When there
exists a splitting $I = I_1+I_2$ of the toric ideal, we show that in some cases
we can describe the (multi-)graded Betti numbers of $I$ in terms of the
(multi-)graded Betti numbers of $I_1$ and $I_2$.
