On PlanetMath.Org regarding the Sum and Product of Functions we find the following:

Let A be a set and K a field or skew field. If f: A $\displaystyle \rightarrow $ K and g:A $\displaystyle \rightarrow $ K, then one can define the product of functions f and g as the function fg: A $\displaystyle \rightarrow $ K as follows:

(fg)(x) := f(x) . g(x) for all x belonging to A

My question is: Why cannot K be a group or a ring with . being the product concerned. Why does K have to be a field or a skew field?

Bernhard