I am stuck with a question that should be very simple, because ALL the books and references that I have read say the same: ''is obvious and we will not prove it''.
Is the following statement: ''Let $ P $ a right $ R $-Module. Prove that if exists an epimorphism $ P^n \longrightarrow R $ for some integer $ n $, then $ P $ is a generator of the category Mod$ -R $.''
At first it seems simple: if $ N $ is a right $ R $-Module and $ S \subset N $ is a generator of $ N $, then $ N $ is isomorphic to a quotient $ R^{(S)} / K $ and so exists an epimorphism $ f : R^{(S)} \longrightarrow N $. On the other hand, we can extend the epimorphism $ g : P^n \longrightarrow R $ given in the statemente to an epimorphism $ \overline{g} : (P^n)^{(S)} \longrightarrow R^{(S)} $ by $$ \overline{g}((x_s)_{s \in S}) = (g(x_s))_{s \in S} $$
Then we obtain an epimorphism $$ f \circ \overline{g} : (P^n)^{(S)} \longrightarrow N $$
That we need is to find an epimorphism $ P^{(I)} \longrightarrow N $ for some set $ I $. That is not exactly the obtained above.
I don't know how to solve this little problem.
Hint: prove that $(P^n)^{(S)}\cong (P^{(S)})^n$