My question is probably stupid and I'm likely committing a very trivial mistake. It's well known that the uniqueness of decomposition of modules into indecomposable submodules ${}_A M = \bigoplus_{I \in I} {}_ A M_i = \bigoplus_{j \in J} {}_ A N_j$ (even when $I$ and $J$ are finite) might fail when the endomorphisms algebras of the respective indecomposables are not local algebras.
I will be concerned with $I$ and $J$ finite here. $A$ might be any $k$-algebra such that $k$ is commutative with unity. Let $1_M = \sum_{i \in I} e_i = \sum_{j \in J} f_j$ be the decompositions into primitive orthogonal idempotents (inside $\text{End}_A ({}_A M)$) associated to, respectively, the decompositions into the $M_i$'s and $N_j$'s.
Notice that
$$e_i = 1_M e_i = \sum_k f_k e_i$$
and, then,
$$e_i (M) = \bigoplus_k f_k (e_i (M)) .$$
Hence $e_i (M) = f_je_i (M)$ for some $j \in J$ since $f_k (M) \cap f_l (M) = 0$ whenever $k \neq l$. Analogously, $f_j (M) = e_k f_j (M)$ for some $k \in I$.
The first equality gives $e_i (M) = f_j (e_i (M)) \subset f_j (M)$. The second gives $f_j (M) = e_k (f_j (M)) \subset e_k (M)$ and, therefore, $e_i (M) \subset e_k (M)$. It follows that $i=k$ since $e_k (M) \cap e_i (M) = 0$ whenever $i\neq k$. Therefore
$$M_i = e_i (M) = f_j (M) = N_j .$$
Now I can take $M/M_i = M/N_j$ and use induction. What's wrong?
Usually, an analogous argument is used to show that finite decompositions of algebras into indecomposable algebras are unique. In that case, however, the fact that the idempotents are central seems to play an essential role. In my above modified case, on the other side, no centrality of the idempotents was assumed.
Extra Questions: Is it possible to find $A$ with two non-isomorphic decompositions of ${}_A A$ into a finite number of indecomposable left ideals (not necessarily bilateral)? Also, will any such decompositions have the same (finite) number of indecomposable summands?
Thanks in advance.