This is exercise 3.4.3 in Etingof's Tensor Categories. Let A be a based ring with basis $\{b_i\}_{i\in I}$ and anti-involution $x\mapsto x^*$. Suppose $M$ is a indecomposable $\mathbb{Z}_+$ module over $A$ with basis $\{m_j\}_{j\in J}$. Prove: $M$ is irreducible (as $\mathbb{Z}_+$ module). In another words, For any subset $J'\subsetneqq \{m_j|j\in J\}$, $\operatorname{span}_\mathbb{Z}(J')$ is not an $A$-submodule.
I try to prove that if $\operatorname{span}_\mathbb{Z}(J')$ is an $A$-submodule, then $\operatorname{span}_\mathbb{Z}(J-J')$ is also an $A$-submodule, which is contradict with the indecomposablity of $M$.
Writing $b_ib_j=\sum_{k\in I}c_{i,j}^kb_k$ and $b_im_j=\sum_{k\in J}a_{i,j}^km_k$. I find that I need an equality like $a_{i,j}^k=a_{i^*,k}^j,\ \forall i\in I, \forall j,k\in J$. But I cannot prove it, neither am I sure if this is true.