Non-finitely generated module isomorphism theorems

Question

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.

Answer 1

I am not sure about your first statement. Let $p$ be a prime, and consider the $\mathbb{Z}$-module $$ A = \mathbb{Z} / p^{2} \mathbb{Z} \oplus \mathbb{Z} / p \mathbb{Z} \oplus \mathbb{Z} / p \mathbb{Z} \oplus \mathbb{Z} / p \mathbb{Z} \oplus \dots. $$ It does not look like $A \cong A \oplus A$.

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Then, if you mean that two automorphisms $\alpha, \beta$ are conjugated if there is another automorphism $\gamma$ such that $\beta = \gamma \alpha \gamma^{-1}$, then I don't think the second statement is valid either.

Note first that if $F$ is the set of fixed point of $\alpha$, then $\gamma(F)$ is the set of fixed points of $\beta$.

Consider $A$ to be the vector space over $\mathbb{Z} / p \mathbb{Z}$ with countable basis $e_{1}, e_{2}, \dots$. Take $\varphi : e_{i} \to e_{i} + e_{i+1}$. This has no non-trivial fixed points, and the same holds true for $\varphi \oplus \varphi$ on $A \oplus A$. So $\varphi \oplus \varphi$ cannot be conjugated to $\varphi \oplus \operatorname{id}_{A}$, which has plenty of fixed points.