I have module theory mid-term exam and I studied on modules over the ring I solve many questions but I reached my main problem is to prove the well-definition. in this question, the goal is proving M is End(M)-left module, I proved End(M) is a ring and M is an abelian group so just remind to prove well-definition of map from End(M)*M to M st. for all f in End(M) and for all m in M, fm=f(m), is there any fixed solution of proving well defined?
End(M) is a set of all M to M homomorphisms
Proving that $M$ is a well-defined $\mathrm{End}(M)$-module really only requires defining the action. How does $\mathrm{End}(M)$ act on $M$? And does it satisfy the axioms of a module action? $$ f.(m+n) = f.m+f.n \\ (f+g).m = f.m + g.m \\ (fg).m = f.(g.m) $$
For contrast, a case where you might need to do more work to prove an action is well-defined is if you're dealing with a quotient. For example, suppose that $N$ is a submodule of $M$, regarding both as left $M$-modules. Then the quotient $M/N$ is a left $M$-module too: for $\overline{x} \in M/N$ you define the action as $m.\overline{x} = \overline{m.x}$. But this choice depends on the representative $x \in M$ you chose of the element in the quotient $\overline{x} \in M/N$. Does this definition of the action still work if you pick a different representative? I.e. if $\overline{x} = \overline{y}$, then does $\overline{m.x} = \overline{m.y}$? In this case, asking if the action is "well-defined" is more meaningful.