Gradings and direct sums notation

Question

I'm studying gradings of rings (specifically polynomial rings) and keep coming across notation looking like this:

$$\bigoplus_{\rho\in\mathcal{I}} {D_{\rho}}\mathbb{Z}$$

in this situation ${D_{\rho}}$ is a divisor. My question is what does this summand actually mean. I see that it can be isomorphic to several coppies of the integers but what do the product mean in this situation.

another example is

$$\bigoplus_{\rho\in\mathcal{I}} {x_{\rho}}\mathbb{C},$$

where $x_{\rho}$ is a monomial, and I am similarly confused.

Answer 1

''another example is $\bigoplus_{\rho\in\mathcal{I}} {x_{\rho}}\mathbb{C}$''

Each element of $\bigoplus_{i\in I} \Bbb C x_i$ is a finite sum $\sum_{i\in I} c_ix_i$, where $c_i\in\Bbb C$.