On $0$-semisimplicity of linear hulls of generators for semigroups generated by idempotents

Let $I$ be a finite set (without $0$) and $J$ a subset of $I\times I$ without diagonal elements. Let $S(I,J)$ denotes the semigroup generated by $e_0=0$ and $e_i$, $i\in I$, with the following relations: $e_i^2=e_i$ for any $i\in I$, $e_ie_j=0$ for any $(i,j)\in J$. In this paper we prove that, for any finite semigroup $S=S(I,J)$ and any its matrix representation $M$ over a field $k$, each matrix of the form $\sum_{i \in I}\alpha_i M(e_i)$ with $\alpha_i\in k$ is similar to the direct sum of some invertible and zero matrices. We also formulate this fact in terms of elements of the semigroup algebra.