Exercise 14, § 2

Let A be a ring, M an A-module, and N a primordial direct factor of M. Let $M =\bigoplus_{i\in I}M_i$ be a decomposition of M into a direct sum, and let $(p_i)_{i\in I}$ be the associated family of projectors. Prove that there exists an $i\in I$ such that $p_i$ induces an isomorphism from N to a direct factor of $M_i$.