I would like to know if the following chain of arguments is correct, since I have a feeling that it is somehow "too nice" to be true. The post might seem long but it should be easy to debunk for the experts here.
Let $\Lambda$ be a finite-dimensional algebra and $M$ a non-injective right $\Lambda$-module. Let $E(M)$ be the injective hull of $M$ and assume that the cosyzygy of $M$ is of the form $I\oplus X$, where $I$ is some injective indecomposable $\Lambda$-module, that is the following is a short exact sequence
$$ 0 \rightarrow M \hookrightarrow E(M) \twoheadrightarrow I\oplus X \rightarrow 0. $$
If $M=\bigoplus_{i=1}^{k}M_i$ is a direct sum decomposition of $M$, then the above sequence can be written
$$ 0 \rightarrow \bigoplus_{i=1}^{k}M_i \overset{\phi}{\hookrightarrow} \bigoplus_{i=1}^{k}E(M_i) \twoheadrightarrow I\oplus X \rightarrow 0$$
where $I\oplus X \cong \text{coker} \phi$ and $\phi=[\phi_{ij}]$ with $\phi_{ij}=0$ if and only if $i\neq j$ (this is one of the parts where I'm not sure this is correct). Assuming this is ok, then $I$ appears as the direct summand of one of $\text{coker}\phi_i$ for $1\leq i \leq k$, let's say without loss of generality $\text{coker} \phi_1 \cong I\oplus X'$ for some $\Lambda$-module $X'$. Then the following is a short exact sequence
$$0\rightarrow M_1\overset{\phi_1}{\hookrightarrow} E(M_1) \twoheadrightarrow I\oplus X'\rightarrow 0.$$
The last claim that I want to make is that if $Y$ is an indecomposable summand of $E(M_1)$, then $\text{Hom}_{\Lambda}(Y,I)\neq 0$. This is the other part that I am not sure, but it should follow from minimality of $\phi_1$.
Finally, if all of this is correct, is the dual statement for the projectives true (with every step of the arguments "dualized")? That is, if $P(N)$ is the projective cover of a non-projective module $N$, and if $\Omega N$ has a projective indecomposabe direct summand $P$, then there exists some indecomposable summand $N_1$ of $N$ such that
$$ 0\rightarrow \Omega N_1 \cong P\oplus Z \hookrightarrow P(N_1) \twoheadrightarrow N_1 \rightarrow 0 $$
is a short exact sequence and if $Y$ is an indecomposable direct summand of $P(N_1)$, then $\text{Hom}_{\Lambda}(P,Y)\neq 0$.