# Thread: Sum and Product of Functions

1. ## Sum and Product of Functions

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

2. ## Re: Sum and Product of Functions

Originally Posted by Bernhard
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?
More general, we only need a binary operation $\displaystyle * :K\times K\to K$ i.e. for all $\displaystyle k,s\in K$, $\displaystyle k* s\in K$ . So we can define $\displaystyle f*g:A\to K$ in the way $\displaystyle (f* g)(x):=f(x)* g(x)$ .