I'm searching for nice isomorphism theorems for non-finitely generated $R$-modules.
I guess that if $A$ is an $R$-module which is not finitely generated, then there is an isomorphism between $A\oplus A$ and $A$.
I really hope it works, I'm able to prove this for countable generated modules, but I don't know how to prove it in general.
Another question is,
if I have an automorphism $\varphi:A\to A$ is then $\varphi\oplus \varphi: A\oplus A\to A\oplus A$ an isomorphism which is conjugated to $\varphi \oplus \text{id}_A$? [Like above $A$ is not finitely generated.]
This question is motivated by the definition of $K_1$ as universal determined and so on..., because in the defintion it is restricted to finitely generated projective $R$-modules, but I think it is no problem if we allow $A$ to be a not finitely generated projective $R$-module, because the above statement/conjecture is true.
Thanks for your help.