 Solution

Generators define a surjection $R^k\to M$, and conversely images of the standard basis generate any quotient. Writing $\varphi (m_i)={\sum }_j{a}_{ij}{m}_j$ with $a_{ij}\in I$ and applying the adjugate to $XI-A$ gives a monic annihilating polynomial whose lower coefficients lie in $I$. Taking $\varphi =1$ when $IM=M$ yields $xM=0$ with $x-1\in I$; for $M=\mathbb{Z}/3$ and $I=(2)$ take $x=3$. The Jacobson radical criterion follows by placing a nonunit $1-rs$ in a maximal ideal. It yields Nakayama’s lemma. Applying the determinant trick to preimages of a finite generating set constructs a polynomial right inverse to any surjective endomorphism, proving injectivity.