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I was reviewing my homework and it seems I overlooked something crucial while proving some ring has no Invariant Basis Number property. This is exercise VI.1.12 in Aluffi's Algebra: Chapter 0

The setup: $V$ is a $k$-vector space and let $R = mathrm{End}_{k}(V)$.

1. Prove that $mathrm{End}_{k}(Voplus V) cong R^4$ as an $R$-module


2. Prove that $R$ doesn't satisfy the IBN property if $V = k^{oplus mathbb N}$.

For the first, I used to the fact that $V oplus V$ is both the product and coproduct (in $k$-Vect) of $V$ with itself to get the isomorphism. What I just realized is I only showed that the two are isomorphic as groups not $R$-modules. So what would be the $R$-module structure on $mathrm{End}_{k}(V oplus V)$?

For the second, I used the fact that $V = k^{oplus mathbb N}$ implies $V cong V oplus V$ which in turn implies $R = mathrm{End}_{k}(V) cong mathrm{End}_{k}(V oplus V)$. Again, I just realized that I only showed the latter two are isomorphic as groups.

It may be obvious (and maybe why my professor let it pass?) but I can't come up with a good $R$-module structure that makes the two group isomorphisms $R$-linear.

Edit:

Explicitly, these are the isomorphisms I'm dealing with. Let $pi_j, i_j$ be the natural projection/inclusion maps of the $j$-th factor resp. and $psi: k^{oplus mathbb N} oplus k^{oplus mathbb N} to k^{oplus mathbb N}$ the isomorphism given by $psi(e_i, 0)=e_{2i-1}$ and $psi(0, e_i)=e_{2i}$.

Then the first isomorphism $mathrm{End}_k(V oplus V)to R^4$ is given by $varphi mapsto (pi_1varphi i_1,pi_2varphi i_1,pi_1varphi i_2,pi_2varphi i_2)$

The second isomorphism $R to mathrm{End}_k(V oplus V)$ is given by $alpha mapsto psi^{-1} alpha psi$

The composition doesn't seem to be $R$-linear if I use the obvious $R$-module structure on $R$ and $R^4$.

I know how to prove that $R$ does not satisfy IBN (still thinking about the first question). Take a basis ${e_imid iinmathbb{N}}$ for $V$ as a $k$-vector space. Define $f_1,f_2in R$ by $f_1(e_i)=e_{2i-1}$ and $f_2(e_i)=e_{2i}$. Then ${f_1,f_2}$ generates $R$ as a right $R$-module, and this set is $R$-linearly independent. So $R^{2}$ and $R$ are isomorphic as $R$-modules, because ${1}$ is also a basis for $R$ as $R$-module.

The way the question is posed, it seems implied that we can use the first part to prove the second. But that is impossible as far I can tell: the induced composition $R cong R^4$ is not $R$-linear.

However, the first isomorphism can be made $R$-linear by using the following structure:

$alpha in R, varphi in mathrm{End}_k(V oplus V) $ then $$ alpha cdot varphi = (alpha oplus alpha) circ varphi $$

This structure appears to be lost through any isomorphism $R cong mathrm{End}_k(V oplus V)$.