Suppose you take the free $\mathbb Z$-module with two generators $\{P,N\}$ and quotient out by the relation $P+N = 0$. You get a one-dimensional module that has two additively independent elements $P$ and $-P$. By additive independence I mean you cannot write any one element as a linear combination of the others with positive coefficients. The two independent elements correspond to the two types of electromagnetic charge, positive and negative.
If instead we start with three generators $\{R,B,G\}$ and quotient out by $R+B+G$ we get a structure that models the three types of color charge and their negatives for quarks. Like everyone knows, each quark can have red, blue, green, antired, antiblue, or antigreen charge. But they only occur in pairs of three, one of each color, which compose a particle with neutral color charge.
Here the elements $-R,-B,-G$ are the antired, antiblue, and antigreen charges. The relation red + blue = antigreen comes out of the relation $R+B+G= 0$ when we subtract $G$ from both sides. This time there are six additively independent elements.
In general suppose we start with the module generated by $\{g_1,\ldots, g_n\}$ and quotient out by $g_1+\ldots+ g_n=0$. What is the additive dimension of the quotient module?